Introduction
Let L L L K K K S L = K [ x α : α ∈ L ] S_L = K[x_\alpha: \alpha\in L] S L = K [ x α : α ∈ L ] L L L join-meet ideal I L I_L I L L L L S L S_L S L x α x β − x α ∧ β x α ∨ β x_\alpha x_\beta- x_{\alpha\wedge\beta}x_{\alpha\vee\beta} x α x β − x α ∧ β x α ∨ β α , β ∈ L \alpha,\beta\in L α , β ∈ L [9 ] , who proved that K [ L ] = S L / I L K[L] = S_L/I_L K [ L ] = S L / I L L L L [6 , 8 ] and the references therein.
In this paper we focus on the thin distributive lattices, also known as generalized snake posets [12 ] . A finite distributive lattice L L L m + 1 m+1 m + 1 i i i i ∈ [ m ] : = { 1 , … , m } i \in[m] := \{1,\ldots,m\} i ∈ [ m ] := { 1 , … , m } c i c_i c i d i d_i d i A ⊆ [ m ] A \subseteq[m] A ⊆ [ m ]
J A = I L + ( x c i − x d i : i ∈ A ) ⊆ S L . J_A = I_L + (x_{c_i} - x_{d_i} : i \in A) \subseteq S_L. J A = I L + ( x c i − x d i : i ∈ A ) ⊆ S L . and we ask when J A J_A J A turning ranks of L L L 2 ≤ k ≤ m − 1 2 \leq k \leq m-1 2 ≤ k ≤ m − 1 c k − 1 ∨ d k − 1 = c k + 1 ∧ d k + 1 c_{k-1} \vee d_{k-1} = c_{k+1} \wedge d_{k+1} c k − 1 ∨ d k − 1 = c k + 1 ∧ d k + 1 Theorem 2.4 , states that J A J_A J A A A A k k k A A A k − 1 k-1 k − 1 k + 1 k+1 k + 1 J A J_A J A S L / J A S_L/J_A S L / J A A ⊆ [ m ] A \subseteq[m] A ⊆ [ m ] A A A S L / J A S_L/J_A S L / J A K [ L ] K[L] K [ L ] [1 ] , where it is shown that, in suitable coordinates, such a model is the toric variety of the Hibi ring of L L L
2020 Mathematics Subject Classification. 06D05, 13C70, 13H10, 13P10.
Key words and phrases. Distributive lattice, join-meet ideal, Hibi ring, prime ideal, lattice ideal, Cohen–Macaulay ring, linear resolution.
This research was supported through the program “Oberwolfach Research Fellows” (OWRF) by the Mathematics Forschungsinstitut Oberwolfach in 2026. The second and third authors are supported by Scientific and Technological Research Council of Turkey TÜBİTAK under the Grants 126F175 and 124F113, respectively; they are thankful to TÜBİTAK for their support. The second author is a member of GNSAGA Indam and he acknowledges their support. results then say that the homogeneous relations among the probabilities of the resulting submodel are generated by the join-meet relations together with the obvious linear ones if and only if A A A J A J_A J A
The paper is organized as follows. In Section 2 we fix the terminology and describe the structure of thin distributive lattices: they are obtained by gluing diamonds and, roughly speaking, they look like staircases whose steps are products of a chain and a chain with two elements. In Section 3 we prove that the two conditions above are necessary, by exhibiting explicit zero-divisors modulo J A J_A J A J A J_A J A Λ A \Lambda_A Λ A Λ A \Lambda_A Λ A L L L Λ A \Lambda_A Λ A A A A
Thin distributive lattices and rank identifications
Posets and distributive lattices
Posets and distributive lattices. A partially ordered set , or poset , is a set P P P ≤ \le ≤ p , q ∈ P p,q \in P p , q ∈ P comparable if p ≤ q p \le q p ≤ q q ≤ p q \le p q ≤ p incomparable otherwise.
A chain of a finite poset P P P P P P length is its cardinality minus one. The rank of P P P rank ( P ) \operatorname{rank}(P) rank ( P ) P P P r ≥ 1 r \ge1 r ≥ 1 C r C_r C r { 0 < 1 < ⋯ < r − 1 } \{0 < 1 < \cdots< r-1\} { 0 < 1 < ⋯ < r − 1 } P P P Q Q Q P × Q P \times Q P × Q P P P Q Q Q ( p , q ) ≤ ( p ′ , q ′ ) (p,q) \le(p',q') ( p , q ) ≤ ( p ′ , q ′ ) p ≤ p ′ p \le p' p ≤ p ′ P P P q ≤ q ′ q \le q' q ≤ q ′ Q Q Q U U U P P P poset ideal if q ∈ U q \in U q ∈ U q ≤ p q \le p q ≤ p p ∈ U p \in U p ∈ U J ( P ) J(P) J ( P ) P P P
A finite lattice is a finite poset L L L α , β \alpha,\beta α , β α ∧ β \alpha\wedge\beta α ∧ β α ∨ β \alpha\vee\beta α ∨ β meet and the join of α \alpha α β \beta β distributive if meet and join distribute over each other. We denote by 0 L 0_L 0 L 1 L 1_L 1 L L L L α ≤ β \alpha\le\beta α ≤ β [ α , β ] = { γ ∈ L : α ≤ γ ≤ β } [\alpha,\beta]=\{\gamma\in L:\alpha\le\gamma\le\beta\} [ α , β ] = { γ ∈ L : α ≤ γ ≤ β } α < β \alpha< \beta α < β [ α , β ] = { α , β } [\alpha,\beta]=\{\alpha,\beta\} [ α , β ] = { α , β } β \beta β covers α \alpha α { α , β } \{\alpha,\beta\} { α , β } edge of L L L L L L diamond is a poset isomorphic to C 2 × C 2 C_2 \times C_2 C 2 × C 2 { a , c , d , b } \{a,c,d,b\} { a , c , d , b } a < c < b a<c<b a < c < b a < d < b a<d<b a < d < b c , d c,d c , d lower edges { a , c } \{a,c\} { a , c } { a , d } \{a,d\} { a , d } upper edges { c , b } \{c,b\} { c , b } { d , b } \{d,b\} { d , b } adjacent if they share an element, and opposite otherwise. An element α ∈ L \alpha\in L α ∈ L apex of L L L L L L L L L simple if its only apexes are 0 L 0_L 0 L 1 L 1_L 1 L 0 L ≠ p ∈ L 0_L \ne p \in L 0 L = p ∈ L join-irreducible if p = α ∨ β p=\alpha\vee\beta p = α ∨ β p = α p=\alpha p = α p = β p=\beta p = β P L P_L P L L L L α ∈ L \alpha\in L α ∈ L
i α = { p ∈ P L : p ≤ α } . \mathfrak{i}_\alpha=\{p \in P_L:p \le\alpha\}. i α = { p ∈ P L : p ≤ α } . By Birkhoff’s fundamental structure theorem [6 Theorem 9.1.7] , if L L L α ↦ i α \alpha\mapsto\mathfrak{i}_\alpha α ↦ i α L L L J ( P L ) J(P_L) J ( P L ) i α ∧ β = i α ∩ i β \mathfrak{i}_{\alpha\wedge\beta}=\mathfrak{i}_\alpha\cap\mathfrak{i}_\beta i α ∧ β = i α ∩ i β i α ∨ β = i α ∪ i β \mathfrak{i}_{\alpha\vee\beta}=\mathfrak{i}_\alpha\cup\mathfrak{i}_\beta i α ∨ β = i α ∪ i β α , β ∈ L \alpha,\beta\in L α , β ∈ L rank of α ∈ L \alpha\in L α ∈ L rank L ( α ) \operatorname{rank}_L(\alpha) rank L ( α ) L L L α \alpha α rank ( L ) = rank L ( 1 L ) \operatorname{rank}(L)=\operatorname{rank}_L(1_L) rank ( L ) = rank L ( 1 L ) [11 Chapter 3] ).
Remark 2.1. Let L L L
(a) rank L ( α ) = ∣ i α ∣ \operatorname{rank}_L(\alpha)=|\mathfrak{i}_\alpha| rank L ( α ) = ∣ i α ∣ α ∈ L \alpha\in L α ∈ L α < β \alpha< \beta α < β rank L ( α ) < rank L ( β ) \operatorname{rank}_L(\alpha)<\operatorname{rank}_L(\beta) rank L ( α ) < rank L ( β ) [ α , β ] [\alpha,\beta] [ α , β ] rank L ( β ) − rank L ( α ) \operatorname{rank}_L(\beta)-\operatorname{rank}_L(\alpha) rank L ( β ) − rank L ( α ) L L L α ≤ β \alpha\le\beta α ≤ β [ α , β ] [\alpha,\beta] [ α , β ] ρ \rho ρ rank L ( α ) ≤ ρ ≤ rank L ( β ) \operatorname{rank}_{L}(\alpha) \le\rho\le\operatorname{rank}_{L}(\beta) rank L ( α ) ≤ ρ ≤ rank L ( β )
(b) rank L ( α ∧ β ) + rank L ( α ∨ β ) = rank L ( α ) + rank L ( β ) \operatorname{rank}_{L}(\alpha\wedge\beta) + \operatorname{rank}_{L}(\alpha\vee\beta) = \operatorname{rank}_{L}(\alpha) + \operatorname{rank}_{L}(\beta) rank L ( α ∧ β ) + rank L ( α ∨ β ) = rank L ( α ) + rank L ( β ) α , β ∈ L \alpha,\beta\in L α , β ∈ L
(c) For all α ≤ β \alpha\le\beta α ≤ β [ α , β ] [\alpha,\beta] [ α , β ] L L L
Thin distributive lattices
For a finite distributive lattice L L L i ≥ 0 i \ge0 i ≥ 0 ρ L ( i ) \rho_{L}(i) ρ L ( i ) L L L i i i L L L d d d thin if ρ L ( i ) = 2 \rho_{L}(i) = 2 ρ L ( i ) = 2 1 ≤ i ≤ d − 1 1 \le i \le d - 1 1 ≤ i ≤ d − 1 [7 ] . Thin distributive lattices are also called generalized snake posets [12 ] . Their structure is easily read off from the definition. We describe it in the following discussion, which also fixes the notation used throughout the paper.
Discussion 2.2. From now on, unless otherwise stated, L L L m + 1 m + 1 m + 1 m ≥ 1 m \ge1 m ≥ 1 L = { 0 L , 1 L } ∪ ˙ { c i , d i : i ∈ [ m ] } L = \{0_{L},1_{L}\} \mathbin{\dot\cup} \{c_{i},d_{i}: i \in[m]\} L = { 0 L , 1 L } ∪ ˙ { c i , d i : i ∈ [ m ]} c i c_{i} c i d i d_{i} d i L L L i i i ∣ L ∣ = 2 m + 2 \lvert L\rvert= 2m + 2 ∣ L ∣ = 2 m + 2 i ∈ [ m ] i \in[m] i ∈ [ m ] a i = c i ∧ d i a_{i} = c_{i} \wedge d_{i} a i = c i ∧ d i b i = c i ∨ d i b_{i} = c_{i} \vee d_{i} b i = c i ∨ d i D i = [ a i , b i ] D_{i} = [a_{i},b_{i}] D i = [ a i , b i ] elementary diamond of rank i i i
The elementary diamonds are the building blocks of L L L i ∈ [ m ] i \in[m] i ∈ [ m ] D i D_{i} D i D i = { a i , c i , d i , b i } D_{i} = \{a_{i},c_{i},d_{i},b_{i}\} D i = { a i , c i , d i , b i } rank L ( a i ) = i − 1 \operatorname{rank}_{L}(a_{i}) = i - 1 rank L ( a i ) = i − 1 rank L ( b i ) = i + 1 \operatorname{rank}_{L}(b_{i}) = i + 1 rank L ( b i ) = i + 1 a 1 = 0 L a_{1} = 0_{L} a 1 = 0 L b m = 1 L b_{m} = 1_{L} b m = 1 L L = D 1 ∪ ⋯ ∪ D m L = D_{1} \cup\cdots\cup D_{m} L = D 1 ∪ ⋯ ∪ D m 1 ≤ i ≤ m − 1 1 \le i \le m - 1 1 ≤ i ≤ m − 1 a i + 1 ∈ { c i , d i } a_{i+1} \in\{c_{i},d_{i}\} a i + 1 ∈ { c i , d i } b i ∈ { c i + 1 , d i + 1 } b_{i} \in\{c_{i+1},d_{i+1}\} b i ∈ { c i + 1 , d i + 1 } D i ∩ D i + 1 = { a i + 1 , b i } D_{i} \cap D_{i+1} = \{a_{i+1},b_{i}\} D i ∩ D i + 1 = { a i + 1 , b i } D i D_{i} D i D i + 1 D_{i+1} D i + 1 L L L D 1 , … , D m D_{1},\ldots,D_{m} D 1 , … , D m Figure 1 .
Figure 1. The Hasse diagram of a thin distributive lattice of rank 13. The elementary diamonds of turning rank are shaded, so that T ( L ) = { 3 , 5 , 6 , 7 , 9 , 10 , 11 } \mathcal{T}(L)=\{3,5,6,7,9,10,11\} T ( L ) = { 3 , 5 , 6 , 7 , 9 , 10 , 11 }
Let 2 ≤ k ≤ m − 1 2 \le k \le m - 1 2 ≤ k ≤ m − 1 D k D_{k} D k { a k , b k − 1 } \{a_{k},b_{k-1}\} { a k , b k − 1 } D k − 1 D_{k-1} D k − 1 { a k + 1 , b k } \{a_{k+1},b_{k}\} { a k + 1 , b k } D k + 1 D_{k+1} D k + 1 b k − 1 , a k + 1 ∈ { c k , d k } b_{k-1},a_{k+1} \in\{c_{k},d_{k}\} b k − 1 , a k + 1 ∈ { c k , d k } b k − 1 ≠ a k + 1 b_{k-1} \ne a_{k+1} b k − 1 = a k + 1 D k D_{k} D k D k − 1 ∩ D k + 1 = ∅ D_{k-1} \cap D_{k+1} = \varnothing D k − 1 ∩ D k + 1 = ∅ b k − 1 = a k + 1 b_{k-1} = a_{k+1} b k − 1 = a k + 1 D k − 1 ∩ D k + 1 = { b k − 1 } D_{k-1} \cap D_{k+1} = \{b_{k-1}\} D k − 1 ∩ D k + 1 = { b k − 1 } L L L k k k k ∈ { 2 , … , m − 1 } k \in\{2,\ldots,m - 1\} k ∈ { 2 , … , m − 1 } turning rank of L L L b k − 1 = a k + 1 b_{k-1} = a_{k+1} b k − 1 = a k + 1 c k − 1 ∨ d k − 1 = c k + 1 ∧ d k + 1 c_{k-1} \vee d_{k-1} = c_{k+1} \wedge d_{k+1} c k − 1 ∨ d k − 1 = c k + 1 ∧ d k + 1 T ( L ) \mathcal{T}(L) T ( L ) L L L T ( L ) = { 3 , 5 , 6 , 7 , 9 , 10 , 11 } \mathcal{T}(L) = \{3,5,6,7,9,10,11\} T ( L ) = { 3 , 5 , 6 , 7 , 9 , 10 , 11 } Figure 1 .
An interval of L L L C N + 1 × C 2 C_{N+1} \times C_{2} C N + 1 × C 2 N ≥ 1 N \ge1 N ≥ 1 ladder . Consecutive elementary diamonds with no turning rank in between are glued along opposite edges, and so they form a ladder. We fix the following notation for ladders, which will be used in several proofs.
Let $1 \le s \le t \le m$ be such that no turning rank k k k L L L s ≤ k < t s \le k < t s ≤ k < t N = t − s + 1 N = t - s + 1 N = t − s + 1 [ a s , b t ] = D s ∪ ⋯ ∪ D t [a_{s},b_{t}] = D_{s} \cup\cdots\cup D_{t} [ a s , b t ] = D s ∪ ⋯ ∪ D t λ j \lambda_{j} λ j μ j \mu_{j} μ j [ a s , b t ] [a_{s},b_{t}] [ a s , b t ] ( j , 0 ) (j,0) ( j , 0 ) ( j , 1 ) (j,1) ( j , 1 ) C N + 1 × C 2 C_{N+1} \times C_{2} C N + 1 × C 2 Figure 2 . Thus λ j = a s + j \lambda_{j} = a_{s+j} λ j = a s + j 0 ≤ j ≤ N − 1 0 \le j \le N - 1 0 ≤ j ≤ N − 1 μ j = b s + j − 1 \mu_{j} = b_{s+j-1} μ j = b s + j − 1 1 ≤ j ≤ N 1 \le j \le N 1 ≤ j ≤ N j ∈ [ N ] j \in[N] j ∈ [ N ] D s + j − 1 = [ λ j − 1 , μ j − 1 ] = { λ j − 1 , μ j − 1 , λ j , μ j } D_{s+j-1} = [\lambda_{j-1},\mu_{j-1}] = \{\lambda_{j-1},\mu_{j-1},\lambda_{j},\mu_{j}\} D s + j − 1 = [ λ j − 1 , μ j − 1 ] = { λ j − 1 , μ j − 1 , λ j , μ j } { c s + j − 1 , d s + j − 1 } = { μ j − 1 , λ j } \{c_{s+j-1},d_{s+j-1}\} = \{\mu_{j-1},\lambda_{j}\} { c s + j − 1 , d s + j − 1 } = { μ j − 1 , λ j } C N + 1 × C 2 C_{N+1} \times C_{2} C N + 1 × C 2 λ i ≤ μ j \lambda_{i} \le\mu_{j} λ i ≤ μ j i ≤ j i \le j i ≤ j μ i ≰ λ j \mu_{i} \not\le\lambda_{j} μ i ≤ λ j 0 ≤ i , j ≤ N 0 \le i,j \le N 0 ≤ i , j ≤ N 0 ≤ i < j ≤ N 0 \le i < j \le N 0 ≤ i < j ≤ N μ i \mu_{i} μ i λ j \lambda_{j} λ j μ i ∧ λ j = λ i \mu_{i} \wedge\lambda_{j} = \lambda_{i} μ i ∧ λ j = λ i μ i ∨ λ j = μ j \mu_{i} \vee\lambda_{j} = \mu_{j} μ i ∨ λ j = μ j
Figure 2. The Hasse diagram of a ladder C 6 × C 2 C_6 \times C_2 C 6 × C 2
Finally, let k 1 < ⋯ < k q k_{1} < \cdots< k_{q} k 1 < ⋯ < k q L L L k 0 = 1 k_{0} = 1 k 0 = 1 k q + 1 = m k_{q+1} = m k q + 1 = m [ a k j , b k j + 1 ] [a_{k_{j}},b_{k_{j+1}}] [ a k j , b k j + 1 ] 0 ≤ j ≤ q 0 \le j \le q 0 ≤ j ≤ q D k j D_{k_{j}} D k j L L L L L L Figure 1 . Since, for k ∈ T ( L ) k \in\mathcal{T}(L) k ∈ T ( L ) b k − 1 = a k + 1 b_{k-1} = a_{k+1} b k − 1 = a k + 1 L L L L L L μ i \mu_{i} μ i λ j \lambda_{j} λ j i < j i < j i < j
Join-meet ideals and rank identifications
Let K K K S L = K [ x α : α ∈ L ] S_{L} = K[x_{\alpha} : \alpha\in L] S L = K [ x α : α ∈ L ] K K K L L L deg x α = 1 \deg x_{\alpha} = 1 deg x α = 1 N = { 0 , 1 , 2 , … } \mathbb{N} = \{0,1,2,\ldots\} N = { 0 , 1 , 2 , … } u ∈ N L \mathbf{u} \in\mathbb{N}^{L} u ∈ N L u ( w ) \mathbf{u}(w) u ( w ) u \mathbf{u} u w ∈ L w \in L w ∈ L x u = ∏ w ∈ L x w u ( w ) \mathbf{x}^{\mathbf{u}}=\prod_{w\in L}x_w^{u(w)} x u = ∏ w ∈ L x w u ( w ) S L S_L S L x u \mathbf{x}^{\mathbf{u}} x u u ∈ N L \mathbf{u}\in\mathbb{N}^{L} u ∈ N L binomial we mean a polynomial x u − x v \mathbf{x}^{\mathbf{u}}-\mathbf{x}^{\mathbf{v}} x u − x v u , v ∈ N L \mathbf{u},\mathbf{v}\in\mathbb{N}^{L} u , v ∈ N L u ≠ v \mathbf{u}\ne\mathbf{v} u = v join-meet ideal of L L L
I L = ( x α x β − x α ∧ β x α ∨ β : α , β ∈ L incomparable ) ⊆ S L , I_L=\left(x_\alpha x_\beta-x_{\alpha\wedge\beta}x_{\alpha\vee\beta}:\alpha,\beta\in L\text{ incomparable}\right)\subseteq S_L, I L = ( x α x β − x α ∧ β x α ∨ β : α , β ∈ L incomparable ) ⊆ S L , and its generators are called the join-meet binomials of L L L [9 ] , the Hibi ring K [ L ] = S L / I L K[L]=S_L/I_L K [ L ] = S L / I L ∣ P L ∣ + 1 |P_L|+1 ∣ P L ∣ + 1 x α 1 x α 2 ⋯ x α q x_{\alpha_1}x_{\alpha_2}\cdots x_{\alpha_q} x α 1 x α 2 ⋯ x α q α 1 ≤ α 2 ≤ ⋯ ≤ α q \alpha_1\le\alpha_2\le\cdots\le\alpha_q α 1 ≤ α 2 ≤ ⋯ ≤ α q K K K K [ L ] K[L] K [ L ] ∣ P L ∣ = rank L ( 1 L ) = m + 1 |P_L|=\operatorname{rank}_L(1_L)=m+1 ∣ P L ∣ = rank L ( 1 L ) = m + 1 dim K [ L ] = m + 2 \dim K[L]=m+2 dim K [ L ] = m + 2
For every i ∈ [ m ] i\in[m] i ∈ [ m ] ℓ i = x c i − x d i \ell_i=x_{c_i}-x_{d_i} ℓ i = x c i − x d i A ⊆ [ m ] A\subseteq[m] A ⊆ [ m ]
J A = I L + ( ℓ i : i ∈ A ) . J_A=I_L+(\ell_i:i\in A). J A = I L + ( ℓ i : i ∈ A ) . Thus S L / J A S_L/J_A S L / J A K [ L ] K[L] K [ L ] i i i i ∈ A i\in A i ∈ A J A J_A J A i i i c i c_i c i d i d_i d i ℓ i \ell_i ℓ i − ℓ i -\ell_i − ℓ i
Definition 2.3. A subset A ⊆ [ m ] A \subseteq[m] A ⊆ [ m ] admissible for L L L
(i) A ∩ T ( L ) = ∅ A \cap T(L) = \varnothing A ∩ T ( L ) = ∅
(ii) { k − 1 , k + 1 } ⊈ A \{k-1,k+1\} \not\subseteq A { k − 1 , k + 1 } ⊆ A k ∈ T ( L ) k \in T(L) k ∈ T ( L )
Our main result is the following.
Theorem 2.4. Let L L L m + 1 m+1 m + 1 A ⊆ [ m ] A \subseteq[m] A ⊆ [ m ] J A J_A J A A A A
Necessary conditions and the associated lattice ideal
In this section we prove that the conditions of Definition 2.3 are necessary for the primality of J A J_A J A J A J_A J A L L L Figure 3 .
Figure 3. The elementary diamonds D k − 1 D_{k-1} D k − 1 D k D_k D k D k + 1 D_{k+1} D k + 1 k k k
Setup 3.1. Let k ∈ T ( L ) k \in T(L) k ∈ T ( L ) d = b k − 1 = a k + 1 d=b_{k-1}=a_{k+1} d = b k − 1 = a k + 1 c = a k c=a_k c = a k f = b k f=b_k f = b k a = a k − 1 a=a_{k-1} a = a k − 1 v = b k + 1 v=b_{k+1} v = b k + 1 e e e b b b u u u k k k k − 1 k-1 k − 1 k + 1 k+1 k + 1 d d d c c c f f f D k − 1 = { a , b , c , d } D_{k-1}=\{a,b,c,d\} D k − 1 = { a , b , c , d } D k = { c , d , e , f } D_k=\{c,d,e,f\} D k = { c , d , e , f } D k + 1 = { d , u , f , v } D_{k+1}=\{d,u,f,v\} D k + 1 = { d , u , f , v } { b , c } \{b,c\} { b , c } { d , e } \{d,e\} { d , e } { u , f } \{u,f\} { u , f } { b , e } \{b,e\} { b , e } { u , e } \{u,e\} { u , e } b ∧ c = a b\wedge c=a b ∧ c = a b ∨ c = d b\vee c=d b ∨ c = d d ∧ e = c d\wedge e=c d ∧ e = c d ∨ e = f d\vee e=f d ∨ e = f u ∧ f = d u\wedge f=d u ∧ f = d u ∨ f = v u\vee f=v u ∨ f = v b ∧ e = a b\wedge e=a b ∧ e = a b ∨ e = f b\vee e=f b ∨ e = f u ∧ e = c u\wedge e=c u ∧ e = c u ∨ e = v u\vee e=v u ∨ e = v ℓ k − 1 = ± ( x b − x c ) \ell_{k-1}=\pm(x_b-x_c) ℓ k − 1 = ± ( x b − x c ) ℓ k = ± ( x d − x e ) \ell_k=\pm(x_d-x_e) ℓ k = ± ( x d − x e ) ℓ k + 1 = ± ( x u − x f ) \ell_{k+1}=\pm(x_u-x_f) ℓ k + 1 = ± ( x u − x f ) a = 0 L a=0_L a = 0 L k = 2 k=2 k = 2 v = 1 L v=1_L v = 1 L k = m − 1 k=m-1 k = m − 1
The next lemma provides a tool to show that certain binomials do not belong to J A J_A J A
Lemma 3.2. Let A ⊆ [ m ] A \subseteq[m] A ⊆ [ m ] k ∈ T ( L ) k \in T(L) k ∈ T ( L ) E a = { c k − 2 , d k − 2 } E_a=\{c_{k-2},d_{k-2}\} E a = { c k − 2 , d k − 2 } k − 2 ∈ A k-2 \in A k − 2 ∈ A E a = { a } E_a=\{a\} E a = { a } E v = { c k + 2 , d k + 2 } E_v=\{c_{k+2},d_{k+2}\} E v = { c k + 2 , d k + 2 } k + 2 ∈ A k+2 \in A k + 2 ∈ A E v = { v } E_v=\{v\} E v = { v } N = K [ ε , η ] / ( ε 2 , η 2 ) \mathcal{N}=K[\varepsilon,\eta]/(\varepsilon^2,\eta^2) N = K [ ε , η ] / ( ε 2 , η 2 ) χ : S L → N \chi:S_L\to\mathcal{N} χ : S L → N K K K
χ ( x w ) = { ε , w ∈ E a , η , w ∈ E v , 0 , w ∉ E a ∪ E v . \chi(x_w)=
\begin{cases}
\varepsilon, & w\in E_a,\\
\eta, & w\in E_v,\\
0, & w\notin E_a\cup E_v.
\end{cases} χ ( x w ) = ⎩ ⎨ ⎧ ε , η , 0 , w ∈ E a , w ∈ E v , w ∈ / E a ∪ E v . Then J A ⊆ ker χ J_A\subseteq\ker\chi J A ⊆ ker χ χ ( x a x v ) = ε η ≠ 0 \chi(x_ax_v)=\varepsilon\eta\ne0 χ ( x a x v ) = ε η = 0 χ ( x b x u ) = χ ( x e 2 ) = 0 \chi(x_bx_u)=\chi(x_e^2)=0 χ ( x b x u ) = χ ( x e 2 ) = 0
Proof. All the elements of E a E_a E a k − 2 k-2 k − 2 E v E_v E v k + 2 k+2 k + 2 E a ∩ E v = ∅ E_a\cap E_v=\varnothing E a ∩ E v = ∅ χ \chi χ a ∈ E a a\in E_a a ∈ E a v ∈ E v v\in E_v v ∈ E v a a a k − 2 k-2 k − 2 v v v k + 2 k+2 k + 2 { c k − 2 , d k − 2 } \{c_{k-2},d_{k-2}\} { c k − 2 , d k − 2 } { c k + 2 , d k + 2 } \{c_{k+2},d_{k+2}\} { c k + 2 , d k + 2 } k − 2 ∈ A k-2 \in A k − 2 ∈ A k + 2 ∈ A k+2 \in A k + 2 ∈ A b , u , e b,u,e b , u , e k − 1 k-1 k − 1 k + 1 k+1 k + 1 k k k χ ( x a x v ) = ε η \chi(x_ax_v)=\varepsilon\eta χ ( x a x v ) = ε η χ ( x b x u ) = χ ( x e 2 ) = 0 \chi(x_bx_u)=\chi(x_e^2)=0 χ ( x b x u ) = χ ( x e 2 ) = 0
Let i ∈ A i \in A i ∈ A i ∉ { k − 2 , k + 2 } i \notin\{k-2,k+2\} i ∈ / { k − 2 , k + 2 } c i , d i ∉ E a ∪ E v c_i,d_i \notin E_a \cup E_v c i , d i ∈ / E a ∪ E v χ ( ℓ i ) = 0 \chi(\ell_i)=0 χ ( ℓ i ) = 0 i = k − 2 i=k-2 i = k − 2 c i , d i ∈ E a c_i,d_i \in E_a c i , d i ∈ E a χ ( ℓ i ) = 0 \chi(\ell_i)=0 χ ( ℓ i ) = 0 i = k + 2 i=k+2 i = k + 2 ℓ i ∈ ker χ \ell_i \in\ker\chi ℓ i ∈ ker χ i ∈ A i \in A i ∈ A d = b k − 1 = a k + 1 d=b_{k-1}=a_{k+1} d = b k − 1 = a k + 1 k k k k k k Figure 1 . In particular, every element of E a E_a E a E v E_v E v
Now note that, for w , w ′ ∈ L w,w' \in L w , w ′ ∈ L χ ( x w x w ′ ) ≠ 0 \chi(x_wx_{w'}) \ne0 χ ( x w x w ′ ) = 0 w , w ′ w,w' w , w ′ E a E_a E a E v E_v E v ε 2 = η 2 = 0 \varepsilon^2=\eta^2=0 ε 2 = η 2 = 0 α , β ∈ L \alpha,\beta\in L α , β ∈ L α \alpha α β \beta β E a E_a E a E v E_v E v χ ( x α x β ) = 0 \chi(x_\alpha x_\beta)=0 χ ( x α x β ) = 0 χ ( x α ∧ β x α ∨ β ) ≠ 0 \chi(x_{\alpha\wedge\beta}x_{\alpha\vee\beta}) \ne0 χ ( x α ∧ β x α ∨ β ) = 0 rank L ( α ∧ β ) < rank L ( α ∨ β ) \operatorname{rank}_L(\alpha\wedge\beta)<\operatorname{rank}_L(\alpha\vee\beta) rank L ( α ∧ β ) < rank L ( α ∨ β ) α ∧ β ∈ E a \alpha\wedge\beta\in E_a α ∧ β ∈ E a α ∨ β ∈ E v \alpha\vee\beta\in E_v α ∨ β ∈ E v rank L ( α ∧ β ) = k − 2 \operatorname{rank}_L(\alpha\wedge\beta)=k-2 rank L ( α ∧ β ) = k − 2 rank L ( α ∨ β ) = k + 2 \operatorname{rank}_L(\alpha\vee\beta)=k+2 rank L ( α ∨ β ) = k + 2 α \alpha α β \beta β k − 2 k-2 k − 2 k + 2 k+2 k + 2 rank L ( α ) + rank L ( β ) = 2 k \operatorname{rank}_L(\alpha)+\operatorname{rank}_L(\beta)=2k rank L ( α ) + rank L ( β ) = 2 k α \alpha α β \beta β k k k k − 1 k-1 k − 1 k + 1 k+1 k + 1 { α , β } = { d , e } \{\alpha,\beta\}=\{d,e\} { α , β } = { d , e } α ∧ β = c \alpha\wedge\beta=c α ∧ β = c k − 1 k-1 k − 1 α , β \alpha,\beta α , β { b , c } \{b,c\} { b , c } { u , f } \{u,f\} { u , f } b , c < d < u , f b,c<d<u,f b , c < d < u , f α \alpha α β \beta β χ ( x α x β − x α ∧ β x α ∨ β ) = 0 \chi(x_\alpha x_\beta-x_{\alpha\wedge\beta}x_{\alpha\vee\beta})=0 χ ( x α x β − x α ∧ β x α ∨ β ) = 0 α , β \alpha,\beta α , β J A ⊆ ker χ J_A \subseteq\ker\chi J A ⊆ ker χ
We are ready to prove that the conditions of Definition 2.3 are necessary for the primality of J A J_A J A
Proposition 3.3. Let L L L m + 1 m+1 m + 1 A ⊆ [ m ] A \subseteq[m] A ⊆ [ m ] A ∩ T ( L ) ≠ ∅ A\cap T(L) \ne\varnothing A ∩ T ( L ) = ∅ J A J_A J A
Proof. Choose k ∈ A ∩ T ( L ) k \in A\cap T(L) k ∈ A ∩ T ( L ) g = x a x v − x b x u g=x_ax_v-x_bx_u g = x a x v − x b x u x c g ∈ J A x_cg \in J_A x c g ∈ J A x c x_c x c g g g J A J_A J A x u x e − x c x v x_ux_e-x_cx_v x u x e − x c x v x b x c − x a x d x_bx_c-x_ax_d x b x c − x a x d L L L x d − x e = ± ℓ k ∈ J A x_d-x_e=\pm\ell_k \in J_A x d − x e = ± ℓ k ∈ J A
x c g = x a ( x c x v − x u x e ) + x u ( x a x d − x b x c ) − x a x u ( x d − x e ) , x_cg=x_a(x_cx_v-x_ux_e)+x_u(x_ax_d-x_bx_c)-x_ax_u(x_d-x_e), x c g = x a ( x c x v − x u x e ) + x u ( x a x d − x b x c ) − x a x u ( x d − x e ) , we get x c g ∈ J A x_cg \in J_A x c g ∈ J A
Let ε : S L → K \varepsilon:S_L\to K ε : S L → K K K K ε ( x w ) = 1 \varepsilon(x_w)=1 ε ( x w ) = 1 w ∈ L w\in L w ∈ L J A J_A J A J A ⊆ ker ε J_A\subseteq\ker\varepsilon J A ⊆ ker ε ε ( M ) = 1 \varepsilon(M)=1 ε ( M ) = 1 M M M J A J_A J A x c ∉ J A x_c\notin J_A x c ∈ / J A J A ⊆ ker χ J_A\subseteq\ker\chi J A ⊆ ker χ χ ( g ) = χ ( x a x v ) − χ ( x b x u ) = ε η ≠ 0 \chi(g)=\chi(x_ax_v)-\chi(x_bx_u)=\varepsilon\eta\ne0 χ ( g ) = χ ( x a x v ) − χ ( x b x u ) = ε η = 0 g ∉ J A g\notin J_A g ∈ / J A J A J_A J A
Proposition 3.4. Let L L L m + 1 m+1 m + 1 A ⊆ [ m ] A \subseteq[m] A ⊆ [ m ] k ∈ T ( L ) k\in T(L) k ∈ T ( L ) { k − 1 , k + 1 } ⊆ A \{k-1,k+1\}\subseteq A { k − 1 , k + 1 } ⊆ A J A J_A J A
Proof. Choose k ∈ T ( L ) k\in T(L) k ∈ T ( L ) k − 1 , k + 1 ∈ A k-1,k+1\in A k − 1 , k + 1 ∈ A g = x a x v − x e 2 g=x_ax_v-x_e^2 g = x a x v − x e 2 x u x f − x d x v x_ux_f-x_dx_v x u x f − x d x v x b x e − x a x f x_bx_e-x_ax_f x b x e − x a x f x d x e − x c x f x_dx_e-x_cx_f x d x e − x c x f L L L x b − x c = ± ℓ k − 1 x_b-x_c=\pm\ell_{k-1} x b − x c = ± ℓ k − 1 x u − x f = ± ℓ k + 1 x_u-x_f=\pm\ell_{k+1} x u − x f = ± ℓ k + 1 J A J_A J A
x d g = x a ( x d x v − x u x f ) + x a x f ( x u − x f ) + x f ( x a x f − x b x e ) + x f x e ( x b − x c ) + x e ( x c x f − x d x e ) , x_dg=x_a(x_dx_v-x_ux_f)+x_ax_f(x_u-x_f)+x_f(x_ax_f-x_bx_e)+x_fx_e(x_b-x_c)+x_e(x_cx_f-x_dx_e), x d g = x a ( x d x v − x u x f ) + x a x f ( x u − x f ) + x f ( x a x f − x b x e ) + x f x e ( x b − x c ) + x e ( x c x f − x d x e ) , we get x d g ∈ J A x_dg\in J_A x d g ∈ J A J A J_A J A x d ∉ J A x_d\notin J_A x d ∈ / J A χ ( g ) = χ ( x a x v ) − χ ( x e 2 ) = ε η ≠ 0 \chi(g)=\chi(x_ax_v)-\chi(x_e^2)=\varepsilon\eta\ne0 χ ( g ) = χ ( x a x v ) − χ ( x e 2 ) = ε η = 0 g ∉ J A g\notin J_A g ∈ / J A J A J_A J A
To prove the converse, we associate with J A J_A J A Z L \mathbb{Z}^L Z L L L L Λ A \Lambda_A Λ A { e w : w ∈ L } \{e_w:w\in L\} { e w : w ∈ L } Z L \mathbb{Z}^L Z L i ∈ [ m ] i\in[m] i ∈ [ m ]
η i = e a i + e b i − e c i − e d i , ρ i = e c i − e d i , \eta_i=e_{a_i}+e_{b_i}-e_{c_i}-e_{d_i},\qquad\rho_i=e_{c_i}-e_{d_i}, η i = e a i + e b i − e c i − e d i , ρ i = e c i − e d i , and, for A ⊆ [ m ] A \subseteq[m] A ⊆ [ m ] Λ A \Lambda_A Λ A Z L \mathbb{Z}^L Z L { η i : i ∈ [ m ] } ∪ { ρ i : i ∈ A } \{\eta_i : i \in[m]\} \cup\{\rho_i : i \in A\} { η i : i ∈ [ m ]} ∪ { ρ i : i ∈ A }
Lemma 3.5. For every A ⊆ [ m ] A \subseteq[m] A ⊆ [ m ] F A ⊆ L F_A \subseteq L F A ⊆ L
B A = { η i : i ∈ [ m ] } ∪ { ρ i : i ∈ A } ∪ { e w : w ∈ F A } \mathcal{B}_A = \{\eta_i : i \in[m]\} \cup\{\rho_i : i \in A\} \cup\{e_w : w \in F_A\} B A = { η i : i ∈ [ m ]} ∪ { ρ i : i ∈ A } ∪ { e w : w ∈ F A } is a Z \mathbb{Z} Z Z L \mathbb{Z}^L Z L Λ A \Lambda_A Λ A { η i : i ∈ [ m ] } ∪ { ρ i : i ∈ A } \{\eta_i : i \in[m]\} \cup\{\rho_i : i \in A\} { η i : i ∈ [ m ]} ∪ { ρ i : i ∈ A } Z \mathbb{Z} Z Λ A \Lambda_A Λ A rank Z Λ A = m + ∣ A ∣ \operatorname{rank}_{\mathbb{Z}} \Lambda_A = m + |A| rank Z Λ A = m + ∣ A ∣
Proof. For 2 ≤ i ≤ m 2 \leq i \leq m 2 ≤ i ≤ m Δ i = D i ∖ ( D 1 ∪ ⋯ ∪ D i − 1 ) \Delta_i = D_i \setminus(D_1 \cup\cdots\cup D_{i-1}) Δ i = D i ∖ ( D 1 ∪ ⋯ ∪ D i − 1 ) D j D_j D j j + 1 j+1 j + 1 D 1 ∪ ⋯ ∪ D i − 1 D_1 \cup\cdots\cup D_{i-1} D 1 ∪ ⋯ ∪ D i − 1 i i i i i i b i − 1 b_{i-1} b i − 1 D i = { a i , c i , d i , b i } D_i = \{a_i,c_i,d_i,b_i\} D i = { a i , c i , d i , b i } a i ∈ { c i − 1 , d i − 1 } ⊆ D i − 1 a_i \in\{c_{i-1},d_{i-1}\} \subseteq D_{i-1} a i ∈ { c i − 1 , d i − 1 } ⊆ D i − 1 b i − 1 ∈ { c i , d i } b_{i-1} \in\{c_i,d_i\} b i − 1 ∈ { c i , d i } rank L ( b i ) = i + 1 \operatorname{rank}_L(b_i) = i+1 rank L ( b i ) = i + 1 Δ i = { p i , q i } \Delta_i = \{p_i,q_i\} Δ i = { p i , q i } p i = b i p_i = b_i p i = b i q i q_i q i { c i , d i } \{c_i,d_i\} { c i , d i } b i − 1 b_{i-1} b i − 1 p 1 = a 1 p_1 = a_1 p 1 = a 1 q 1 = c 1 q_1 = c_1 q 1 = c 1 L L L L = D 1 ⊔ Δ 2 ⊔ ⋯ ⊔ Δ m L = D_1 \sqcup\Delta_2 \sqcup\cdots\sqcup\Delta_m L = D 1 ⊔ Δ 2 ⊔ ⋯ ⊔ Δ m
P A = { p i : i ∈ [ m ] } ∪ { q i : i ∈ A } , F A = { b 1 , d 1 } ∪ { q i : i ∈ [ m ] ∖ A } . P_A = \{p_i : i \in[m]\} \cup\{q_i : i \in A\}, \qquad F_A = \{b_1,d_1\} \cup\{q_i : i \in[m] \setminus A\}. P A = { p i : i ∈ [ m ]} ∪ { q i : i ∈ A } , F A = { b 1 , d 1 } ∪ { q i : i ∈ [ m ] ∖ A } . Since D 1 = { p 1 , q 1 , b 1 , d 1 } D_1 = \{p_1,q_1,b_1,d_1\} D 1 = { p 1 , q 1 , b 1 , d 1 } L = P A ⊔ F A L = P_A \sqcup F_A L = P A ⊔ F A ∣ P A ∣ = m + ∣ A ∣ |P_A| = m + |A| ∣ P A ∣ = m + ∣ A ∣ ∣ F A ∣ = m + 2 − ∣ A ∣ |F_A| = m + 2 - |A| ∣ F A ∣ = m + 2 − ∣ A ∣
For every i ∈ [ m ] i \in[m] i ∈ [ m ] p i p_i p i D i D_i D i q i ∈ { c i , d i } q_i \in\{c_i,d_i\} q i ∈ { c i , d i } η i ( p i ) = 1 \eta_i(p_i) = 1 η i ( p i ) = 1 η i ( q i ) = − 1 \eta_i(q_i) = -1 η i ( q i ) = − 1 ρ i ( p i ) = 0 \rho_i(p_i) = 0 ρ i ( p i ) = 0 ρ i ( q i ) = κ i \rho_i(q_i) = \kappa_i ρ i ( q i ) = κ i κ i = 1 \kappa_i = 1 κ i = 1 q i = c i q_i = c_i q i = c i κ i = − 1 \kappa_i = -1 κ i = − 1 q i = d i q_i = d_i q i = d i j > i j > i j > i p j , q j ∈ Δ j p_j,q_j \in\Delta_j p j , q j ∈ Δ j D i D_i D i η i ( p j ) = η i ( q j ) = ρ i ( p j ) = ρ i ( q j ) = 0 \eta_i(p_j) = \eta_i(q_j) = \rho_i(p_j) = \rho_i(q_j) = 0 η i ( p j ) = η i ( q j ) = ρ i ( p j ) = ρ i ( q j ) = 0
Order the generators of Λ A \Lambda_A Λ A η i \eta_i η i i ∈ A i \in A i ∈ A ρ i \rho_i ρ i i = 1 , … , m i = 1,\ldots,m i = 1 , … , m P A P_A P A p i p_i p i i ∈ A i \in A i ∈ A q i q_i q i i = 1 , … , m i = 1,\ldots,m i = 1 , … , m M A M_A M A ( m + ∣ A ∣ ) × ( m + ∣ A ∣ ) (m+|A|) \times(m+|A|) ( m + ∣ A ∣ ) × ( m + ∣ A ∣ ) Λ A \Lambda_A Λ A P A P_A P A M A M_A M A i i i 1 × 1 1 \times1 1 × 1 ( η i ( p i ) ) = ( 1 ) (\eta_i(p_i)) = (1) ( η i ( p i )) = ( 1 ) i ∉ A i \notin A i ∈ / A
( η i ( p i ) ρ i ( p i ) η i ( q i ) ρ i ( q i ) ) = ( 1 0 − 1 κ i ) \begin{pmatrix}
\eta_i(p_i) & \rho_i(p_i) \\
\eta_i(q_i) & \rho_i(q_i)
\end{pmatrix}
=
\begin{pmatrix}
1 & 0 \\
-1 & \kappa_i
\end{pmatrix} ( η i ( p i ) η i ( q i ) ρ i ( p i ) ρ i ( q i ) ) = ( 1 − 1 0 κ i ) if i ∈ A i \in A i ∈ A det M A = ∏ i ∈ A κ i ∈ { 1 , − 1 } \det M_A = \prod_{i \in A} \kappa_i \in\{1,-1\} det M A = ∏ i ∈ A κ i ∈ { 1 , − 1 }
Now fix an ordering F A = { w 1 , … , w m + 2 − ∣ A ∣ } F_A = \{w_1,\ldots,w_{m+2-|A|}\} F A = { w 1 , … , w m + 2 − ∣ A ∣ } B A \mathcal{B}_A B A Λ A \Lambda_A Λ A e w 1 , … , e w m + 2 − ∣ A ∣ e_{w_1},\ldots,e_{w_{m+2-|A|}} e w 1 , … , e w m + 2 − ∣ A ∣ P A P_A P A w 1 , … , w m + 2 − ∣ A ∣ w_1,\ldots,w_{m+2-|A|} w 1 , … , w m + 2 − ∣ A ∣
( M A 0 N A I ) \begin{pmatrix}
M_A & 0 \\
N_A & I
\end{pmatrix} ( M A N A 0 I ) where N A N_A N A I I I det M A ∈ { 1 , − 1 } \det M_A \in\{1,-1\} det M A ∈ { 1 , − 1 } B A \mathcal{B}_A B A Z \mathbb{Z} Z Z L \mathbb{Z}^L Z L Z L = Λ A ⊕ ⨁ w ∈ F A Z e w \mathbb{Z}^L = \Lambda_A \oplus\bigoplus_{w \in F_A} \mathbb{Z}e_w Z L = Λ A ⊕ ⨁ w ∈ F A Z e w Λ A \Lambda_A Λ A Z L \mathbb{Z}^L Z L
For a subgroup Λ ⊆ Z L \Lambda\subseteq\mathbb{Z}^L Λ ⊆ Z L lattice ideal of Λ \Lambda Λ
I Λ = ( x z + − x z − : z ∈ Λ ) ⊆ S L . I_\Lambda= (x^{z^+} - x^{z^-} : z \in\Lambda) \subseteq S_L. I Λ = ( x z + − x z − : z ∈ Λ ) ⊆ S L . where z + , z − ∈ N L z^+, z^- \in\mathbb{N}^L z + , z − ∈ N L z z z z = z + − z − z = z^+ - z^- z = z + − z − u , v ∈ N L u,v \in\mathbb{N}^L u , v ∈ N L x u − x v x^u - x^v x u − x v I Λ I_\Lambda I Λ u − v ∈ Λ u-v \in\Lambda u − v ∈ Λ I Λ I_\Lambda I Λ Λ \Lambda Λ [8 ] . Since all the generators of Λ A \Lambda_A Λ A I Λ A I_{\Lambda_A} I Λ A I Λ A I_{\Lambda_A} I Λ A
For an ideal Q ⊆ S L Q \subseteq S_L Q ⊆ S L f ∈ S L f \in S_L f ∈ S L Q : f ∞ = ⋃ n ≥ 0 ( Q : f n ) Q:f^{\infty}=\bigcup_{n\geq0}(Q:f^n) Q : f ∞ = ⋃ n ≥ 0 ( Q : f n ) Q : f n = { g ∈ S L : f n g ∈ Q } Q:f^n=\{g\in S_L:f^ng\in Q\} Q : f n = { g ∈ S L : f n g ∈ Q } X L = ∏ w ∈ L x w X_L=\prod_{w\in L}x_w X L = ∏ w ∈ L x w
Proposition 3.6. Let L L L m + 1 m+1 m + 1 A ⊆ [ m ] A\subseteq[m] A ⊆ [ m ] J A ⊆ I Λ A J_A\subseteq I_{\Lambda_A} J A ⊆ I Λ A
I Λ A = J A : X L ∞ . I_{\Lambda_A}=J_A:X_L^{\infty}. I Λ A = J A : X L ∞ . In particular, there exists an integer N ≥ 0 N\geq0 N ≥ 0 I Λ A = J A : X L N I_{\Lambda_A}=J_A:X_L^N I Λ A = J A : X L N
Proof. Consider the ideal
I A b a s = ( x a i x b i − x c i x d i : i ∈ [ m ] ) + ( ℓ i : i ∈ A ) . I_A^{\mathrm{bas}}=(x_{a_i}x_{b_i}-x_{c_i}x_{d_i}:i\in[m])+(\ell_i:i\in A). I A bas = ( x a i x b i − x c i x d i : i ∈ [ m ]) + ( ℓ i : i ∈ A ) . The exponent vectors of its generators are, up to sign, the elements η i \eta_i η i i ∈ [ m ] i\in[m] i ∈ [ m ] ρ i \rho_i ρ i i ∈ A i\in A i ∈ A Z \mathbb{Z} Z Λ A \Lambda_A Λ A I A b a s I_A^{\mathrm{bas}} I A bas Λ A \Lambda_A Λ A [10 Proposition 1.1] gives I Λ A = I A b a s : X L ∞ I_{\Lambda_A}=I_A^{\mathrm{bas}}:X_L^{\infty} I Λ A = I A bas : X L ∞
Since x c i x d i − x a i x b i x_{c_i}x_{d_i}-x_{a_i}x_{b_i} x c i x d i − x a i x b i L L L i ∈ [ m ] i\in[m] i ∈ [ m ] I A b a s ⊆ J A I_A^{\mathrm{bas}}\subseteq J_A I A bas ⊆ J A J A ⊆ I Λ A J_A\subseteq I_{\Lambda_A} J A ⊆ I Λ A ℓ i \ell_i ℓ i ρ i ∈ Λ A \rho_i\in\Lambda_A ρ i ∈ Λ A α , β ∈ L \alpha,\beta\in L α , β ∈ L α \alpha α β \beta β [ a s , b t ] [a_s,b_t] [ a s , b t ] α = μ i \alpha=\mu_i α = μ i β = λ j \beta=\lambda_j β = λ j 0 ≤ i < j ≤ N 0\leq i<j\leq N 0 ≤ i < j ≤ N α ∧ β = λ i \alpha\wedge\beta=\lambda_i α ∧ β = λ i α ∨ β = μ j \alpha\vee\beta=\mu_j α ∨ β = μ j Figure 2 . In other words, the join-meet binomial x α x β − x α ∧ β x α ∨ β x_\alpha x_\beta-x_{\alpha\wedge\beta}x_{\alpha\vee\beta} x α x β − x α ∧ β x α ∨ β D s + i , … , D s + j − 1 D_{s+i},\ldots,D_{s+j-1} D s + i , … , D s + j − 1 η \eta η y ∈ [ N ] y\in[N] y ∈ [ N ] D s + y − 1 = { λ y − 1 , μ y − 1 , λ y , μ y } D_{s+y-1}=\{\lambda_{y-1},\mu_{y-1},\lambda_y,\mu_y\} D s + y − 1 = { λ y − 1 , μ y − 1 , λ y , μ y } λ y − 1 \lambda_{y-1} λ y − 1 μ y \mu_y μ y η s + y − 1 = e λ y − 1 + e μ y − e μ y − 1 − e λ y \eta_{s+y-1}=e_{\lambda_{y-1}}+e_{\mu_y}-e_{\mu_{y-1}}-e_{\lambda_y} η s + y − 1 = e λ y − 1 + e μ y − e μ y − 1 − e λ y i < y ≤ j i<y\leq j i < y ≤ j
e α ∧ β + e α ∨ β − e α − e β = e λ i + e μ j − e μ i − e λ j = ∑ y = i + 1 j η s + y − 1 ∈ Λ A . e_{\alpha\wedge\beta}+e_{\alpha\vee\beta}-e_\alpha-e_\beta=e_{\lambda_i}+e_{\mu_j}-e_{\mu_i}-e_{\lambda_j}=\sum_{y=i+1}^{j}\eta_{s+y-1}\in\Lambda_A. e α ∧ β + e α ∨ β − e α − e β = e λ i + e μ j − e μ i − e λ j = y = i + 1 ∑ j η s + y − 1 ∈ Λ A . Saturating with respect to X L X_L X L
I Λ A = I A b a s : X L ∞ ⊆ J A : X L ∞ ⊆ I Λ A : X L ∞ = I Λ A , I_{\Lambda_A}=I_A^{\mathrm{bas}}:X_L^{\infty}\subseteq J_A:X_L^{\infty}\subseteq I_{\Lambda_A}:X_L^{\infty}=I_{\Lambda_A}, I Λ A = I A bas : X L ∞ ⊆ J A : X L ∞ ⊆ I Λ A : X L ∞ = I Λ A , where the last equality holds because I Λ A : X L ∞ = ( I A b a s : X L ∞ ) : X L ∞ = I A b a s : X L ∞ I_{\Lambda_A}:X_L^{\infty}=(I_A^{\mathrm{bas}}:X_L^{\infty}):X_L^{\infty}=I_A^{\mathrm{bas}}:X_L^{\infty} I Λ A : X L ∞ = ( I A bas : X L ∞ ) : X L ∞ = I A bas : X L ∞ I Λ A = J A : X L ∞ I_{\Lambda_A}=J_A:X_L^{\infty} I Λ A = J A : X L ∞ S L S_L S L J A : X L ⊆ J A : X L 2 ⊆ ⋯ J_A:X_L\subseteq J_A:X_L^2\subseteq\cdots J A : X L ⊆ J A : X L 2 ⊆ ⋯ J A : X L ∞ = J A : X L N J_A:X_L^{\infty}=J_A:X_L^N J A : X L ∞ = J A : X L N N ≥ 0 N\geq0 N ≥ 0
Theorem 3.7. Let L L L m + 1 m+1 m + 1 A ⊆ [ m ] A\subseteq[m] A ⊆ [ m ]
(i) J A J_A J A
(ii) J A = I Λ A J_A=I_{\Lambda_A} J A = I Λ A
(iii) J A : X L ∞ = J A J_A:X_L^{\infty}=J_A J A : X L ∞ = J A
Proof. (ii)⇔ \Leftrightarrow ⇔
(i)⇒ \Rightarrow ⇒ Λ A \Lambda_A Λ A I Λ A I_{\Lambda_A} I Λ A
(i)⇒ \Rightarrow ⇒ J A J_A J A X L ∉ J A X_L\notin J_A X L ∈ / J A J A J_A J A J A : X L n = J A J_A:X_L^n=J_A J A : X L n = J A n ≥ 0 n\geq0 n ≥ 0 J A : X L ∞ = J A J_A:X_L^{\infty}=J_A J A : X L ∞ = J A
:::
Proof of the sufficiency
Throughout this section, L L L m + 1 m+1 m + 1 A ⊆ [ m ] A\subseteq[m] A ⊆ [ m ] J A J_A J A A A A J A = I Λ A J_A=I_{\Lambda_A} J A = I Λ A A A A
A binomial f ∈ I Λ A f \in I_{\Lambda_A} f ∈ I Λ A redundant if it belongs to the ideal generated by the binomials of I Λ A I_{\Lambda_A} I Λ A deg ( f ) \deg(f) deg ( f ) irredundant otherwise. The key result of this section is the following.
Proposition 4.1. Let L L L m + 1 m+1 m + 1 A ⊆ [ m ] A \subseteq[m] A ⊆ [ m ]
(i) every binomial of degree one in I Λ A I_{\Lambda_A} I Λ A ( ℓ i : i ∈ A ) (\ell_i : i \in A) ( ℓ i : i ∈ A ) I Λ A I_{\Lambda_A} I Λ A J A J_A J A
(ii) every irredundant binomial in I Λ A I_{\Lambda_A} I Λ A
The main tool to control the binomials of I Λ A I_{\Lambda_A} I Λ A
Lemma 4.2. Let θ : L → Z \theta: L \to\mathbb{Z} θ : L → Z θ ( a i ) + θ ( b i ) = θ ( c i ) + θ ( d i ) \theta(a_i) + \theta(b_i) = \theta(c_i) + \theta(d_i) θ ( a i ) + θ ( b i ) = θ ( c i ) + θ ( d i ) i ∈ [ m ] i \in[m] i ∈ [ m ] θ ( c i ) = θ ( d i ) \theta(c_i) = \theta(d_i) θ ( c i ) = θ ( d i ) i ∈ A i \in A i ∈ A u ∈ N L \mathbf{u} \in\mathbb{N}^{L} u ∈ N L θ ( u ) = ∑ w ∈ L u ( w ) θ ( w ) \theta(\mathbf{u}) = \sum_{w \in L} u(w)\theta(w) θ ( u ) = ∑ w ∈ L u ( w ) θ ( w ) x u − x v ∈ I Λ A \mathbf{x}^{\mathbf{u}} - \mathbf{x}^{\mathbf{v}} \in I_{\Lambda_A} x u − x v ∈ I Λ A θ ( u ) = θ ( v ) \theta(\mathbf{u}) = \theta(\mathbf{v}) θ ( u ) = θ ( v )
Proof. Let Θ : Z L → Z \Theta: \mathbb{Z}^{L} \to\mathbb{Z} Θ : Z L → Z Θ ( e w ) = θ ( w ) \Theta(\mathbf{e}_w) = \theta(w) Θ ( e w ) = θ ( w ) w ∈ L w \in L w ∈ L Θ ( u ) = θ ( u ) \Theta(\mathbf{u}) = \theta(\mathbf{u}) Θ ( u ) = θ ( u ) u ∈ N L \mathbf{u} \in\mathbb{N}^{L} u ∈ N L i ∈ [ m ] i \in[m] i ∈ [ m ] θ \theta θ Θ ( η i ) = θ ( a i ) + θ ( b i ) − θ ( c i ) − θ ( d i ) = 0 \Theta(\boldsymbol{\eta}_i) = \theta(a_i) + \theta(b_i) - \theta(c_i) - \theta(d_i) = 0 Θ ( η i ) = θ ( a i ) + θ ( b i ) − θ ( c i ) − θ ( d i ) = 0 i ∈ A i \in A i ∈ A Θ ( ρ i ) = θ ( c i ) − θ ( d i ) = 0 \Theta(\boldsymbol{\rho}_i) = \theta(c_i) - \theta(d_i) = 0 Θ ( ρ i ) = θ ( c i ) − θ ( d i ) = 0 Λ A \Lambda_A Λ A Λ A ⊆ ker Θ \Lambda_A \subseteq\ker\Theta Λ A ⊆ ker Θ x u − x v ∈ I Λ A \mathbf{x}^{\mathbf{u}} - \mathbf{x}^{\mathbf{v}} \in I_{\Lambda_A} x u − x v ∈ I Λ A u − v ∈ Λ A \mathbf{u} - \mathbf{v} \in\Lambda_A u − v ∈ Λ A θ ( u ) − θ ( v ) = Θ ( u − v ) = 0 \theta(\mathbf{u}) - \theta(\mathbf{v}) = \Theta(\mathbf{u} - \mathbf{v}) = 0 θ ( u ) − θ ( v ) = Θ ( u − v ) = 0 □ \square □
A function θ \theta θ A A A θ \theta θ A A A i ∈ A i \in A i ∈ A
Let f f f I Λ A I_{\Lambda_A} I Λ A d d d f f f good if one of the following holds:
d = 1 d = 1 d = 1 f ∈ ( ℓ i : i ∈ A ) f \in(\ell_i : i \in A) f ∈ ( ℓ i : i ∈ A )
d = 2 d = 2 d = 2 f ∈ J A f \in J_A f ∈ J A
d ≥ 3 d \ge3 d ≥ 3 f f f
We also agree that 0 0 0 I Λ A I_{\Lambda_A} I Λ A [3 Lemma 2.2] .
Lemma 4.3. Let f = x u − x v ∈ I Λ A f = \mathbf{x}^{\mathbf{u}} - \mathbf{x}^{\mathbf{v}} \in I_{\Lambda_A} f = x u − x v ∈ I Λ A d d d
(a) Let g = x r − x s g = \mathbf{x}^{\mathbf{r}} - \mathbf{x}^{\mathbf{s}} g = x r − x s g = ± ℓ i g = \pm\ell_i g = ± ℓ i i ∈ A i \in A i ∈ A d ≥ 2 d \ge2 d ≥ 2 g g g L L L x r \mathbf{x}^{\mathbf{r}} x r x u \mathbf{x}^{\mathbf{u}} x u u ′ = u − r + s \mathbf{u}' = \mathbf{u} - \mathbf{r} + \mathbf{s} u ′ = u − r + s f ′ = x u ′ − x v f' = \mathbf{x}^{\mathbf{u}'} - \mathbf{x}^{\mathbf{v}} f ′ = x u ′ − x v I Λ A I_{\Lambda_A} I Λ A d d d f f f f ′ f' f ′ x u \mathbf{x}^{\mathbf{u}} x u x v \mathbf{x}^{\mathbf{v}} x v f ′ f' f ′ f f f
(b) If some variable x w x_w x w x u \mathbf{x}^{\mathbf{u}} x u x v \mathbf{x}^{\mathbf{v}} x v f = x w f 1 f = x_w f_1 f = x w f 1 f 1 f_1 f 1 I Λ A I_{\Lambda_A} I Λ A d − 1 d - 1 d − 1 d ≥ 3 d \ge3 d ≥ 3 d = 2 d = 2 d = 2 f 1 f_1 f 1 f f f
Proof. (a) We have f − f ′ = x u − r g ∈ J A ⊆ I Λ A f - f' = \mathbf{x}^{\mathbf{u}-\mathbf{r}}g \in J_A \subseteq I_{\Lambda_A} f − f ′ = x u − r g ∈ J A ⊆ I Λ A f ′ ∈ I Λ A f' \in I_{\Lambda_A} f ′ ∈ I Λ A deg x u ′ = deg x u = d \deg\mathbf{x}^{\mathbf{u}'} = \deg\mathbf{x}^{\mathbf{u}} = d deg x u ′ = deg x u = d g g g d = 1 d = 1 d = 1 g = ± ℓ i g = \pm\ell_i g = ± ℓ i f − f ′ ∈ ( ℓ i : i ∈ A ) f - f' \in(\ell_i : i \in A) f − f ′ ∈ ( ℓ i : i ∈ A ) d = 2 d = 2 d = 2 f − f ′ ∈ J A f - f' \in J_A f − f ′ ∈ J A d ≥ 3 d \ge3 d ≥ 3 f − f ′ f - f' f − f ′ g g g I Λ A I_{\Lambda_A} I Λ A 2 < d 2 < d 2 < d f f f f ′ f' f ′
(b) By Lemma 3.5, Λ A \Lambda_A Λ A I Λ A I_{\Lambda_A} I Λ A I Λ A I_{\Lambda_A} I Λ A S L → K S_L \to K S L → K 1 1 1 1 1 1 x w ∉ I Λ A x_w \notin I_{\Lambda_A} x w ∈ / I Λ A x w f 1 = f ∈ I Λ A x_w f_1 = f \in I_{\Lambda_A} x w f 1 = f ∈ I Λ A f 1 ∈ I Λ A f_1 \in I_{\Lambda_A} f 1 ∈ I Λ A f 1 f_1 f 1 d − 1 d - 1 d − 1 d ≥ 3 d \ge3 d ≥ 3 f f f f 1 ∈ I Λ A f_1 \in I_{\Lambda_A} f 1 ∈ I Λ A d − 1 d - 1 d − 1 d = 2 d = 2 d = 2 f 1 ∈ ( ℓ i : i ∈ A ) f_1 \in(\ell_i : i \in A) f 1 ∈ ( ℓ i : i ∈ A ) f ∈ J A f \in J_A f ∈ J A □ \square □
We now fix the setting in which we perform the induction on the rank of L L L T ( L ) ⊆ { 2 , … , m − 1 } \mathcal{T}(L) \subseteq\{2, \ldots, m - 1\} T ( L ) ⊆ { 2 , … , m − 1 }
Setup 4.4. Let m ≥ 2 m \ge2 m ≥ 2 A \mathcal{A} A r = max T ( L ) r=\max\mathcal{T}(L) r = max T ( L ) T ( L ) ≠ ∅ \mathcal{T}(L)\ne\varnothing T ( L ) = ∅ r = 1 r=1 r = 1 1 ≤ r ≤ m − 1 1\le r\le m-1 1 ≤ r ≤ m − 1 n = m − r + 1 n=m-r+1 n = m − r + 1 n ≥ 2 n\ge2 n ≥ 2 k k k r < k < m r<k<m r < k < m [ a r , b m ] = D r ∪ ⋯ ∪ D m [a_r,b_m]=D_r\cup\cdots\cup D_m [ a r , b m ] = D r ∪ ⋯ ∪ D m α j \alpha_j α j γ j \gamma_j γ j 0 ≤ j ≤ n 0\le j\le n 0 ≤ j ≤ n λ j \lambda_j λ j μ j \mu_j μ j
B = [ a r , 1 L ] = { α 0 , … , α n , γ 0 , … , γ n } ≅ C n + 1 × C 2 , B=[a_r,1_L]=\{\alpha_0,\ldots,\alpha_n,\gamma_0,\ldots,\gamma_n\}\cong C_{n+1}\times C_2, B = [ a r , 1 L ] = { α 0 , … , α n , γ 0 , … , γ n } ≅ C n + 1 × C 2 , α i ≤ γ j \alpha_i\le\gamma_j α i ≤ γ j i ≤ j i\le j i ≤ j γ i ≰ α j \gamma_i\nleq\alpha_j γ i ≰ α j i , j i,j i , j j ∈ [ n ] j\in[n] j ∈ [ n ] B j = D r + j − 1 B_j=D_{r+j-1} B j = D r + j − 1
B j = [ α j − 1 , γ j ] = { α j − 1 , γ j − 1 , α j , γ j } , { c r + j − 1 , d r + j − 1 } = { γ j − 1 , α j } . B_j=[\alpha_{j-1},\gamma_j]=\{\alpha_{j-1},\gamma_{j-1},\alpha_j,\gamma_j\},\qquad\{c_{r+j-1},d_{r+j-1}\}=\{\gamma_{j-1},\alpha_j\}. B j = [ α j − 1 , γ j ] = { α j − 1 , γ j − 1 , α j , γ j } , { c r + j − 1 , d r + j − 1 } = { γ j − 1 , α j } . In particular, the linear binomial attached to B j B_j B j ℓ r + j − 1 = ± ( x γ j − 1 − x α j ) \ell_{r+j-1}=\pm(x_{\gamma_{j-1}}-x_{\alpha_j}) ℓ r + j − 1 = ± ( x γ j − 1 − x α j ) α j = a r + j \alpha_j=a_{r+j} α j = a r + j 0 ≤ j ≤ n − 1 0\le j\le n-1 0 ≤ j ≤ n − 1 γ j = b r + j − 1 \gamma_j=b_{r+j-1} γ j = b r + j − 1 1 ≤ j ≤ n 1\le j\le n 1 ≤ j ≤ n γ n = b m = 1 L \gamma_n=b_m=1_L γ n = b m = 1 L A = { D i : i ∈ A } \mathcal{A}=\{D_i:i\in A\} A = { D i : i ∈ A } A A A B j ∈ A B_j\in\mathcal{A} B j ∈ A r + j − 1 ∈ A r+j-1\in A r + j − 1 ∈ A
We set p = γ n = 1 L p=\gamma_n=1_L p = γ n = 1 L q = α n q=\alpha_n q = α n k 0 k_0 k 0 { 0 , … , n } \{0,\ldots,n\} { 0 , … , n } B j ∈ A B_j\in\mathcal{A} B j ∈ A k 0 < j ≤ n k_0<j\le n k 0 < j ≤ n B k 0 + 1 , … , B n B_{k_0+1},\ldots,B_n B k 0 + 1 , … , B n A \mathcal{A} A B n = D m B_n=D_m B n = D m k 0 = n k_0=n k 0 = n m ∉ A m\notin A m ∈ / A B k 0 ∉ A B_{k_0}\notin\mathcal{A} B k 0 ∈ / A k 0 ≥ 1 k_0\ge1 k 0 ≥ 1
t = n − k 0 , F = { α j , γ j : k 0 ≤ j ≤ n } . t=n-k_0,\qquad F=\{\alpha_j,\gamma_j:k_0\le j\le n\}. t = n − k 0 , F = { α j , γ j : k 0 ≤ j ≤ n } . Thus t ≥ 1 t\ge1 t ≥ 1 m ∈ A m\in A m ∈ A F = { p , q } F=\{p,q\} F = { p , q } m ∉ A m\notin A m ∈ / A Figure 4 .
Figure 4. The setting of Setup 4.4 with n = 6 n=6 n = 6 A \mathcal{A} A k 0 = 3 k_0=3 k 0 = 3 F F F E 1 = D r − 1 E_1=D_{r-1} E 1 = D r − 1 E 2 = D r − 2 E_2=D_{r-2} E 2 = D r − 2
Lemma 4.5. In the setting of Setup 4.4, the following hold.
(S1) The elements p p p q q q D m = B n D_m=B_n D m = B n m m m L ′ = [ 0 L , b m − 1 ] L'=[0_L,b_{m-1}] L ′ = [ 0 L , b m − 1 ] m m m L = L ′ ⊔ { p , q } L=L'\sqcup\{p,q\} L = L ′ ⊔ { p , q } L ′ L' L ′ D 1 , … , D m − 1 D_1,\ldots,D_{m-1} D 1 , … , D m − 1 L ′ L' L ′ L L L T ( L ′ ) ⊆ T ( L ) \mathcal{T}(L')\subseteq\mathcal{T}(L) T ( L ′ ) ⊆ T ( L )
(S2) If r > 1 r>1 r > 1 r ∈ T ( L ) r\in\mathcal{T}(L) r ∈ T ( L ) B 1 ∉ A B_1\notin\mathcal{A} B 1 ∈ / A k 0 ≥ 1 k_0\ge1 k 0 ≥ 1 E 1 = D r − 1 E_1=D_{r-1} E 1 = D r − 1 α 1 \alpha_1 α 1 D i ∩ B ⊆ { α 0 , α 1 } D_i\cap B\subseteq\{\alpha_0,\alpha_1\} D i ∩ B ⊆ { α 0 , α 1 } i < r i<r i < r
(S3) For 0 ≤ k < j ≤ n 0\le k<j\le n 0 ≤ k < j ≤ n γ k \gamma_k γ k α j \alpha_j α j γ k ∧ α j = α k \gamma_k\wedge\alpha_j=\alpha_k γ k ∧ α j = α k γ k ∨ α j = γ j \gamma_k\vee\alpha_j=\gamma_j γ k ∨ α j = γ j g k , j = x α k x γ j − x γ k x α j g_{k,j}=x_{\alpha_k}x_{\gamma_j}-x_{\gamma_k}x_{\alpha_j} g k , j = x α k x γ j − x γ k x α j L L L
Proof. (S1) The elements of D i D_i D i i − 1 i-1 i − 1 i + 1 i+1 i + 1 p = 1 L p=1_L p = 1 L m + 1 m+1 m + 1 D m D_m D m q q q m m m D m − 1 D_{m-1} D m − 1 D m D_m D m D m − 1 D_{m-1} D m − 1 m m m b m − 1 = γ n − 1 b_{m-1}=\gamma_{n-1} b m − 1 = γ n − 1 γ n − 1 ≠ α n = q \gamma_{n-1}\ne\alpha_n=q γ n − 1 = α n = q q q q D m D_m D m
For 1 ≤ i ≤ m − 2 1\leq i\leq m-2 1 ≤ i ≤ m − 2 b i ∈ { c i + 1 , d i + 1 } b_i\in\{c_{i+1},d_{i+1}\} b i ∈ { c i + 1 , d i + 1 } b i ≤ b i + 1 b_i\leq b_{i+1} b i ≤ b i + 1 c i , d i ≤ b i ≤ b m − 1 c_i,d_i\leq b_i\leq b_{m-1} c i , d i ≤ b i ≤ b m − 1 i ∈ [ m − 1 ] i\in[m-1] i ∈ [ m − 1 ] L ′ = [ 0 L , b m − 1 ] L'=[0_L,b_{m-1}] L ′ = [ 0 L , b m − 1 ] 0 L 0_L 0 L 1 , … , m − 1 1,\ldots,m-1 1 , … , m − 1
On the other hand, the only element of L ′ L' L ′ m m m b m − 1 b_{m-1} b m − 1 q q q b m − 1 b_{m-1} b m − 1 1 L ≰ b m − 1 1_L\nleq b_{m-1} 1 L ≰ b m − 1 L = L ′ ⊔ { p , q } L=L'\sqcup\{p,q\} L = L ′ ⊔ { p , q } L ′ L' L ′ L L L rank L \operatorname{rank}_L rank L L ′ L' L ′ m m m i i i i ∈ [ m − 1 ] i\in[m-1] i ∈ [ m − 1 ] m − 1 ≥ 1 m-1\geq1 m − 1 ≥ 1 i i i c i c_i c i d i d_i d i D 1 , … , D m − 1 D_1,\ldots,D_{m-1} D 1 , … , D m − 1 k ∈ { 2 , … , m − 2 } k\in\{2,\ldots,m-2\} k ∈ { 2 , … , m − 2 } L ′ L' L ′ b k − 1 = a k + 1 b_{k-1}=a_{k+1} b k − 1 = a k + 1 k ∈ T ( L ) k\in T(L) k ∈ T ( L ) L ′ L' L ′ L ′ L' L ′ L L L L ′ L' L ′ L L L
(S2) If r > 1 r>1 r > 1 r = max T ( L ) ∈ T ( L ) r=\max T(L)\in T(L) r = max T ( L ) ∈ T ( L ) B 1 = D r ∉ A B_1=D_r\notin\mathcal{A} B 1 = D r ∈ / A k 0 ≥ 1 k_0\geq1 k 0 ≥ 1 r ∈ T ( L ) r\in T(L) r ∈ T ( L ) n ≥ 2 n\geq2 n ≥ 2 D r − 1 D_{r-1} D r − 1 b r − 1 = a r + 1 = α 1 b_{r-1}=a_{r+1}=\alpha_1 b r − 1 = a r + 1 = α 1 i < r i<r i < r D i D_i D i i + 1 ≤ r i+1\leq r i + 1 ≤ r B B B r r r α 0 \alpha_0 α 0 r − 1 r-1 r − 1 γ 0 , α 1 \gamma_0,\alpha_1 γ 0 , α 1 r r r γ 0 ∈ D i \gamma_0\in D_i γ 0 ∈ D i rank L ( γ 0 ) = r \operatorname{rank}_L(\gamma_0)=r rank L ( γ 0 ) = r i = r − 1 i=r-1 i = r − 1 γ 0 = b r − 1 = α 1 \gamma_0=b_{r-1}=\alpha_1 γ 0 = b r − 1 = α 1 D i ∩ B ⊆ { α 0 , α 1 } D_i\cap B\subseteq\{\alpha_0,\alpha_1\} D i ∩ B ⊆ { α 0 , α 1 }
(S3) This follows from Discussion 2.2, since γ k \gamma_k γ k α j \alpha_j α j ( k , 1 ) (k,1) ( k , 1 ) ( j , 0 ) (j,0) ( j , 0 ) C n + 1 × C 2 C_{n+1}\times C_2 C n + 1 × C 2
We introduce two A \mathcal{A} A B B B Figure 4 are displayed in Figure 5 .
Figure 5. The values of the A \mathcal{A} A ω \omega ω ω \omega ω L L L A \mathcal{A} A ω \omega ω
Lemma 4.6. In the setting of Setup 4.4, let ζ ( 0 ) = 0 \zeta(0)=0 ζ ( 0 ) = 0 j ∈ [ n ] j\in[n] j ∈ [ n ] ζ ( j ) = ζ ( j − 1 ) + 1 \zeta(j)=\zeta(j-1)+1 ζ ( j ) = ζ ( j − 1 ) + 1 B j ∈ A B_j\in\mathcal{A} B j ∈ A ζ ( j ) = 0 \zeta(j)=0 ζ ( j ) = 0 ζ ( j ) \zeta(j) ζ ( j ) A \mathcal{A} A B j B_j B j
(a) The function ω : L → Z \omega:L\to\mathbb{Z} ω : L → Z ω ( α j ) = ζ ( j ) \omega(\alpha_j)=\zeta(j) ω ( α j ) = ζ ( j ) ω ( γ j ) = ζ ( j ) + 1 \omega(\gamma_j)=\zeta(j)+1 ω ( γ j ) = ζ ( j ) + 1 0 ≤ j ≤ n 0\leq j\leq n 0 ≤ j ≤ n ω ( w ) = 0 \omega(w)=0 ω ( w ) = 0 w ∉ B w\notin B w ∈ / B A \mathcal{A} A
(b) If k 0 ≠ 1 k_0\ne1 k 0 = 1 r = 1 r=1 r = 1 δ F \delta_F δ F F F F δ F ( w ) = 1 \delta_F(w)=1 δ F ( w ) = 1 w ∈ F w\in F w ∈ F δ F ( w ) = 0 \delta_F(w)=0 δ F ( w ) = 0 A \mathcal{A} A
(c) For k 0 ≤ j ≤ n k_0 \le j \le n k 0 ≤ j ≤ n ω ( α j ) = j − k 0 \omega(\alpha_j)=j-k_0 ω ( α j ) = j − k 0 ω ( γ j ) = j − k 0 + 1 \omega(\gamma_j)=j-k_0+1 ω ( γ j ) = j − k 0 + 1 ω ( p ) = t + 1 \omega(p)=t+1 ω ( p ) = t + 1 ω ( w ) ≤ t \omega(w)\le t ω ( w ) ≤ t w ∈ F ∖ { p } w\in F\setminus\{p\} w ∈ F ∖ { p } ω ( w ) ≥ 1 \omega(w)\ge1 ω ( w ) ≥ 1 w = γ j w=\gamma_j w = γ j 0 ≤ j ≤ n 0\le j\le n 0 ≤ j ≤ n w = α j w=\alpha_j w = α j j ∈ [ n ] j\in[n] j ∈ [ n ] B j ∈ A B_j\in\mathcal{A} B j ∈ A ω ( γ j − 1 ) = ω ( α j ) \omega(\gamma_{j-1})=\omega(\alpha_j) ω ( γ j − 1 ) = ω ( α j )
Proof. (a) Clearly ω ≥ 0 \omega\ge0 ω ≥ 0 j ∈ [ n ] j\in[n] j ∈ [ n ] B j B_j B j α j − 1 \alpha_{j-1} α j − 1 γ j \gamma_j γ j γ j − 1 \gamma_{j-1} γ j − 1 α j \alpha_j α j ω ( α j − 1 ) + ω ( γ j ) \omega(\alpha_{j-1})+\omega(\gamma_j) ω ( α j − 1 ) + ω ( γ j ) ω ( γ j − 1 ) + ω ( α j ) \omega(\gamma_{j-1})+\omega(\alpha_j) ω ( γ j − 1 ) + ω ( α j ) ζ ( j − 1 ) + ζ ( j ) + 1 \zeta(j-1)+\zeta(j)+1 ζ ( j − 1 ) + ζ ( j ) + 1 B j ∈ A B_j\in\mathcal{A} B j ∈ A ω ( γ j − 1 ) = ζ ( j − 1 ) + 1 = ζ ( j ) = ω ( α j ) \omega(\gamma_{j-1})=\zeta(j-1)+1=\zeta(j)=\omega(\alpha_j) ω ( γ j − 1 ) = ζ ( j − 1 ) + 1 = ζ ( j ) = ω ( α j ) r = 1 r=1 r = 1 r > 1 r>1 r > 1 B 1 ∉ A B_1\notin\mathcal{A} B 1 ∈ / A ζ ( 1 ) = 0 \zeta(1)=0 ζ ( 1 ) = 0 ω \omega ω α 0 \alpha_0 α 0 α 1 \alpha_1 α 1 ω \omega ω D i D_i D i i < r i<r i < r
(b) Let j ∈ [ n ] j\in[n] j ∈ [ n ] B j B_j B j F F F j < k 0 j<k_0 j < k 0 F F F j > k 0 j>k_0 j > k 0 γ k 0 \gamma_{k_0} γ k 0 α k 0 \alpha_{k_0} α k 0 F F F j = k 0 ≥ 1 j=k_0\ge1 j = k 0 ≥ 1 δ F \delta_F δ F B j B_j B j B j ∈ A B_j\in\mathcal{A} B j ∈ A j ≠ k 0 j\ne k_0 j = k 0 γ j − 1 \gamma_{j-1} γ j − 1 α j \alpha_j α j F F F F F F r = 1 r=1 r = 1 r > 1 r>1 r > 1 k 0 ≥ 1 k_0\ge1 k 0 ≥ 1 k 0 ≥ 2 k_0\ge2 k 0 ≥ 2 α 0 , α 1 ∉ F \alpha_0,\alpha_1\notin F α 0 , α 1 ∈ / F δ F \delta_F δ F D i D_i D i i < r i<r i < r
(c) We first compute ζ ( j ) \zeta(j) ζ ( j ) k 0 ≤ j ≤ n k_0\le j\le n k 0 ≤ j ≤ n ζ ( k 0 ) = 0 \zeta(k_0)=0 ζ ( k 0 ) = 0 k 0 = 0 k_0=0 k 0 = 0 k 0 ≥ 1 k_0\ge1 k 0 ≥ 1 B k 0 ∉ A B_{k_0}\notin\mathcal{A} B k 0 ∈ / A k 0 k_0 k 0 ζ ( k 0 ) = 0 \zeta(k_0)=0 ζ ( k 0 ) = 0 ζ \zeta ζ k 0 < j ≤ n k_0<j\le n k 0 < j ≤ n B j ∈ A B_j\in\mathcal{A} B j ∈ A ζ ( j ) = ζ ( j − 1 ) + 1 \zeta(j)=\zeta(j-1)+1 ζ ( j ) = ζ ( j − 1 ) + 1 j j j ζ ( j ) = j − k 0 \zeta(j)=j-k_0 ζ ( j ) = j − k 0 k 0 ≤ j ≤ n k_0\le j\le n k 0 ≤ j ≤ n ω ( α j ) = j − k 0 \omega(\alpha_j)=j-k_0 ω ( α j ) = j − k 0 ω ( γ j ) = j − k 0 + 1 \omega(\gamma_j)=j-k_0+1 ω ( γ j ) = j − k 0 + 1 k 0 ≤ j ≤ n k_0\le j\le n k 0 ≤ j ≤ n ω ( p ) = ω ( γ n ) = n − k 0 + 1 = t + 1 \omega(p)=\omega(\gamma_n)=n-k_0+1=t+1 ω ( p ) = ω ( γ n ) = n − k 0 + 1 = t + 1 w ∈ F ∖ { p } w\in F\setminus\{p\} w ∈ F ∖ { p } w = α j w=\alpha_j w = α j k 0 ≤ j ≤ n k_0\le j\le n k 0 ≤ j ≤ n ω ( w ) = j − k 0 ≤ n − k 0 = t \omega(w)=j-k_0\le n-k_0=t ω ( w ) = j − k 0 ≤ n − k 0 = t w = γ j w=\gamma_j w = γ j k 0 ≤ j ≤ n − 1 k_0\le j\le n-1 k 0 ≤ j ≤ n − 1 ω ( w ) = j − k 0 + 1 ≤ n − k 0 = t \omega(w)=j-k_0+1\le n-k_0=t ω ( w ) = j − k 0 + 1 ≤ n − k 0 = t ω ( w ) ≤ t \omega(w)\le t ω ( w ) ≤ t w ∈ F ∖ { p } w\in F\setminus\{p\} w ∈ F ∖ { p }
Finally, let w ∈ L w\in L w ∈ L ω ( w ) ≥ 1 \omega(w)\ge1 ω ( w ) ≥ 1 ω \omega ω B B B w = α j w=\alpha_j w = α j w = γ j w=\gamma_j w = γ j 0 ≤ j ≤ n 0\le j\le n 0 ≤ j ≤ n w = α j w=\alpha_j w = α j ζ ( j ) = ω ( α j ) ≥ 1 \zeta(j)=\omega(\alpha_j)\ge1 ζ ( j ) = ω ( α j ) ≥ 1 ζ ( 0 ) = 0 \zeta(0)=0 ζ ( 0 ) = 0 j ≥ 1 j\ge1 j ≥ 1 ζ ( j ) = 0 \zeta(j)=0 ζ ( j ) = 0 B j ∉ A B_j\notin\mathcal{A} B j ∈ / A B j ∈ A B_j\in\mathcal{A} B j ∈ A ζ ( j ) = ζ ( j − 1 ) + 1 \zeta(j)=\zeta(j-1)+1 ζ ( j ) = ζ ( j − 1 ) + 1 ζ \zeta ζ ω ( γ j − 1 ) = ζ ( j − 1 ) + 1 = ζ ( j ) = ω ( α j ) \omega(\gamma_{j-1})=\zeta(j-1)+1=\zeta(j)=\omega(\alpha_j) ω ( γ j − 1 ) = ζ ( j − 1 ) + 1 = ζ ( j ) = ω ( α j )
The following lemma is the heart of the proof: it shows how to move, inside B B B F F F x p x_p x p
Lemma 4.7. In the setting of Setup 4.4, let f = x u − x v ∈ I Λ f=\mathbf{x}^{\mathbf{u}}-\mathbf{x}^{\mathbf{v}}\in I_{\Lambda} f = x u − x v ∈ I Λ d d d x p ∣ x u x_p\mid\mathbf{x}^{\mathbf{u}} x p ∣ x u x p ∤ x v x_p\nmid\mathbf{x}^{\mathbf{v}} x p ∤ x v x w ∣ x v x_w\mid\mathbf{x}^{\mathbf{v}} x w ∣ x v w ∈ F w\in F w ∈ F I Λ I_{\Lambda} I Λ d d d f f f
Proof. Let a a a ω ( w ) \omega(w) ω ( w ) w ∈ F w\in F w ∈ F x w ∣ x v x_w\mid\mathbf{x}^{\mathbf{v}} x w ∣ x v x p ∣ x v x_p\mid\mathbf{x}^{\mathbf{v}} x p ∣ x v a ≤ t a\le t a ≤ t t − a t-a t − a j = k 0 + a j=k_0+a j = k 0 + a k 0 ≤ j ≤ n k_0\le j\le n k 0 ≤ j ≤ n k 0 ≤ k ≤ n k_0\le k\le n k 0 ≤ k ≤ n ω ( α k ) = k − k 0 \omega(\alpha_k)=k-k_0 ω ( α k ) = k − k 0 k = j k=j k = j ω ( γ k ) = k − k 0 + 1 ≤ a \omega(\gamma_k)=k-k_0+1\le a ω ( γ k ) = k − k 0 + 1 ≤ a k = j − 1 k=j-1 k = j − 1 j − 1 ≥ k 0 j-1\ge k_0 j − 1 ≥ k 0 a ≥ 1 a\ge1 a ≥ 1 F F F ω \omega ω a a a α j \alpha_j α j a ≥ 1 a\ge1 a ≥ 1 γ j − 1 \gamma_{j-1} γ j − 1 a a a x v \mathbf{x}^{\mathbf{v}} x v x α j x_{\alpha_j} x α j x γ j − 1 x_{\gamma_{j-1}} x γ j − 1
We claim that we may assume x α j ∣ x v x_{\alpha_j}\mid\mathbf{x}^{\mathbf{v}} x α j ∣ x v x α j ∤ x v x_{\alpha_j}\nmid\mathbf{x}^{\mathbf{v}} x α j ∤ x v a ≥ 1 a\ge1 a ≥ 1 x γ j − 1 ∣ x v x_{\gamma_{j-1}}\mid\mathbf{x}^{\mathbf{v}} x γ j − 1 ∣ x v j > k 0 j>k_0 j > k 0 B j ∈ A B_j\in\mathcal{A} B j ∈ A B j B_j B j ± ( x γ j − 1 − x α j ) \pm(x_{\gamma_{j-1}}-x_{\alpha_j}) ± ( x γ j − 1 − x α j ) x v \mathbf{x}^{\mathbf{v}} x v g = x γ j − 1 − x α j g=x_{\gamma_{j-1}}-x_{\alpha_j} g = x γ j − 1 − x α j x γ j − 1 x_{\gamma_{j-1}} x γ j − 1 x v \mathbf{x}^{\mathbf{v}} x v x α j x_{\alpha_j} x α j f ′ = x u − x v ′ f'=\mathbf{x}^{\mathbf{u}}-\mathbf{x}^{\mathbf{v}'} f ′ = x u − x v ′ v ′ = v − e γ j − 1 + e α j \mathbf{v}'=\mathbf{v}-\mathbf{e}_{\gamma_{j-1}}+\mathbf{e}_{\alpha_j} v ′ = v − e γ j − 1 + e α j I Λ I_{\Lambda} I Λ d d d f f f f ′ f' f ′ f ′ = 0 f'=0 f ′ = 0 f ′ f' f ′ f ′ f' f ′ f f f a a a x u \mathbf{x}^{\mathbf{u}} x u x p ∤ x v ′ x_p\nmid\mathbf{x}^{\mathbf{v}'} x p ∤ x v ′ α j ≠ p \alpha_j\ne p α j = p x v ′ \mathbf{x}^{\mathbf{v}'} x v ′ x α j x_{\alpha_j} x α j α j ∈ F \alpha_j\in F α j ∈ F F F F x v ′ \mathbf{x}^{\mathbf{v}'} x v ′ x v \mathbf{x}^{\mathbf{v}} x v f f f f ′ f' f ′ x α j ∣ x v x_{\alpha_j}\mid\mathbf{x}^{\mathbf{v}} x α j ∣ x v Lemmas 4.2 and 4.6 (a), we have ω ( v ) = ω ( u ) ≥ ω ( p ) = t + 1 \omega(\mathbf{v})=\omega(\mathbf{u})\geq\omega(p)=t+1 ω ( v ) = ω ( u ) ≥ ω ( p ) = t + 1 x v = x α j x v ′ \mathbf{x}^{\mathbf{v}}=x_{\alpha_j}\mathbf{x}^{\mathbf{v}'} x v = x α j x v ′ ω ( v ′ ) = ω ( v ) − a ≥ t + 1 − a ≥ 1 \omega(\mathbf{v}')=\omega(\mathbf{v})-a\geq t+1-a\geq1 ω ( v ′ ) = ω ( v ) − a ≥ t + 1 − a ≥ 1 d ≥ 2 d\geq2 d ≥ 2 ω \omega ω ω ( v ′ ) = ∑ w ∈ L v ′ ( w ) ω ( w ) ≥ 1 \omega(\mathbf{v}')=\sum_{w\in L}v'(w)\omega(w)\geq1 ω ( v ′ ) = ∑ w ∈ L v ′ ( w ) ω ( w ) ≥ 1 w ′ ∈ L w'\in L w ′ ∈ L x w ′ x_{w'} x w ′ x v ′ \mathbf{x}^{\mathbf{v}'} x v ′ ω ( w ′ ) ≥ 1 \omega(w')\geq1 ω ( w ′ ) ≥ 1 Lemma 4.6 (c), either w ′ = γ k w'=\gamma_k w ′ = γ k 0 ≤ k ≤ n 0\leq k\leq n 0 ≤ k ≤ n w ′ = α k w'=\alpha_k w ′ = α k k ∈ [ n ] k\in[n] k ∈ [ n ] B k ∈ A B_k\in\mathcal{A} B k ∈ A ω ( γ k − 1 ) = ω ( α k ) \omega(\gamma_{k-1})=\omega(\alpha_k) ω ( γ k − 1 ) = ω ( α k )
We show that the second case can be reduced to the first one. Assume that w ′ = α k w'=\alpha_k w ′ = α k B k ∈ A B_k\in\mathcal{A} B k ∈ A Lemma 4.3 (a) with g = x α k − x γ k − 1 g=x_{\alpha_k}-x_{\gamma_{k-1}} g = x α k − x γ k − 1 x α k x_{\alpha_k} x α k x v ′ \mathbf{x}^{\mathbf{v}'} x v ′ x γ k − 1 x_{\gamma_{k-1}} x γ k − 1 f ′ = x u − x α j x v ′ ′ f'=\mathbf{x}^{\mathbf{u}}-x_{\alpha_j}\mathbf{x}^{\mathbf{v}''} f ′ = x u − x α j x v ′′ v ′ ′ = v ′ − e α k + e γ k − 1 \mathbf{v}''=\mathbf{v}'-\mathbf{e}_{\alpha_k}+\mathbf{e}_{\gamma_{k-1}} v ′′ = v ′ − e α k + e γ k − 1 f ′ f' f ′ I Λ A I_{\Lambda_A} I Λ A d d d f f f f ′ f' f ′ f ′ f' f ′ f f f a a a j j j x u \mathbf{x}^{\mathbf{u}} x u x α j x v ′ ′ x_{\alpha_j}\mathbf{x}^{\mathbf{v}''} x α j x v ′′ x α j x_{\alpha_j} x α j x p x_p x p γ k − 1 ≠ p \gamma_{k-1}\ne p γ k − 1 = p α k ∈ F \alpha_k\in F α k ∈ F k ≥ k 0 k\geq k_0 k ≥ k 0 k ≥ 1 k\geq1 k ≥ 1 B k 0 ∉ A B_{k_0}\notin\mathcal{A} B k 0 ∈ / A k 0 ≥ 1 k_0\geq1 k 0 ≥ 1 k > k 0 k>k_0 k > k 0 γ k − 1 ∈ F \gamma_{k-1}\in F γ k − 1 ∈ F ω ( γ k − 1 ) = ω ( α k ) ≤ a \omega(\gamma_{k-1})=\omega(\alpha_k)\leq a ω ( γ k − 1 ) = ω ( α k ) ≤ a α k ∉ F \alpha_k\notin F α k ∈ / F k < k 0 k<k_0 k < k 0 γ k − 1 ∉ F \gamma_{k-1}\notin F γ k − 1 ∈ / F ω \omega ω F F F x α j x v ′ ′ x_{\alpha_j}\mathbf{x}^{\mathbf{v}''} x α j x v ′′ a a a α j \alpha_j α j ω ( v ′ ′ ) = ω ( v ′ ) \omega(\mathbf{v}'')=\omega(\mathbf{v}') ω ( v ′′ ) = ω ( v ′ ) x γ k − 1 x_{\gamma_{k-1}} x γ k − 1 x v ′ ′ \mathbf{x}^{\mathbf{v}''} x v ′′ ω ( γ k − 1 ) ≥ 1 \omega(\gamma_{k-1})\geq1 ω ( γ k − 1 ) ≥ 1 f f f f ′ f' f ′ w ′ = γ k w'=\gamma_k w ′ = γ k 0 ≤ k ≤ n 0\leq k\leq n 0 ≤ k ≤ n
Then k ≤ n − 1 k\leq n-1 k ≤ n − 1 x p ∤ x v x_p\nmid\mathbf{x}^{\mathbf{v}} x p ∤ x v k < j k<j k < j γ k ∉ F \gamma_k\notin F γ k ∈ / F k < k 0 ≤ j k<k_0\leq j k < k 0 ≤ j γ k ∈ F \gamma_k\in F γ k ∈ F k − k 0 + 1 = ω ( γ k ) ≤ a = j − k 0 k-k_0+1=\omega(\gamma_k)\leq a=j-k_0 k − k 0 + 1 = ω ( γ k ) ≤ a = j − k 0 a a a g = x γ k x α j − x α k x γ j = − g k , j g=x_{\gamma_k}x_{\alpha_j}-x_{\alpha_k}x_{\gamma_j}=-g_{k,j} g = x γ k x α j − x α k x γ j = − g k , j L L L x γ k x α j x_{\gamma_k}x_{\alpha_j} x γ k x α j x v = x α j x v ′ \mathbf{x}^{\mathbf{v}}=x_{\alpha_j}\mathbf{x}^{\mathbf{v}'} x v = x α j x v ′ x γ k x_{\gamma_k} x γ k x v ′ \mathbf{x}^{\mathbf{v}'} x v ′ d ≥ 2 d\geq2 d ≥ 2 Lemma 4.3 (a) with this g g g x γ k x α j x_{\gamma_k}x_{\alpha_j} x γ k x α j x α k x γ j x_{\alpha_k}x_{\gamma_j} x α k x γ j x v \mathbf{x}^{\mathbf{v}} x v f ′ = x u − x v ∗ f'=\mathbf{x}^{\mathbf{u}}-\mathbf{x}^{\mathbf{v}^*} f ′ = x u − x v ∗ v ∗ = v − e γ k − e α j + e α k + e γ j \mathbf{v}^*=\mathbf{v}-\mathbf{e}_{\gamma_k}-\mathbf{e}_{\alpha_j}+\mathbf{e}_{\alpha_k}+\mathbf{e}_{\gamma_j} v ∗ = v − e γ k − e α j + e α k + e γ j f ′ f' f ′ I Λ A I_{\Lambda_A} I Λ A d d d f f f f ′ f' f ′ f ′ ≠ 0 f'\ne0 f ′ = 0
Assume first that a = t a=t a = t j = k 0 + t = n j=k_0+t=n j = k 0 + t = n γ j = p \gamma_j=p γ j = p x p x_p x p x u \mathbf{x}^{\mathbf{u}} x u x v ∗ \mathbf{x}^{\mathbf{v}^*} x v ∗ f ′ = x p f 1 f'=x_p f_1 f ′ = x p f 1 f 1 ∈ I Λ A f_1\in I_{\Lambda_A} f 1 ∈ I Λ A d − 1 d-1 d − 1 d ≥ 3 d\geq3 d ≥ 3 f ′ f' f ′ Lemma 4.3 (b). If d = 2 d=2 d = 2 f 1 f_1 f 1 d d d f ′ f' f ′ Lemma 4.3 (b). This settles the case t − a = 0 t-a=0 t − a = 0
Assume now that a < t a<t a < t j < n j<n j < n γ j ≠ p \gamma_j\ne p γ j = p α k ≠ p \alpha_k\ne p α k = p x p ∤ x v x_p\nmid\mathbf{x}^{\mathbf{v}} x p ∤ x v x p ∤ x v ∗ x_p\nmid\mathbf{x}^{\mathbf{v}^*} x p ∤ x v ∗ x v ∗ \mathbf{x}^{\mathbf{v}^*} x v ∗ x γ j x_{\gamma_j} x γ j γ j ∈ F \gamma_j\in F γ j ∈ F ω ( γ j ) = j − k 0 + 1 = a + 1 \omega(\gamma_j)=j-k_0+1=a+1 ω ( γ j ) = j − k 0 + 1 = a + 1 Lemma 4.6 (c). Let w ∈ F ∖ { γ j } w\in F\setminus\{\gamma_j\} w ∈ F ∖ { γ j } x w x_w x w x v ∗ \mathbf{x}^{\mathbf{v}^*} x v ∗ w ≠ α k w\ne\alpha_k w = α k x w x_w x w x v \mathbf{x}^{\mathbf{v}} x v ω ( w ) ≤ a \omega(w)\leq a ω ( w ) ≤ a a a a w = α k w=\alpha_k w = α k γ k ∈ F \gamma_k\in F γ k ∈ F α k \alpha_k α k γ k \gamma_k γ k F F F k ≥ k 0 k\geq k_0 k ≥ k 0 x γ k x_{\gamma_k} x γ k x v \mathbf{x}^{\mathbf{v}} x v ω ( α k ) = ω ( γ k ) − 1 ≤ a − 1 \omega(\alpha_k)=\omega(\gamma_k)-1\leq a-1 ω ( α k ) = ω ( γ k ) − 1 ≤ a − 1 f ′ f' f ′ ω \omega ω F F F x v ∗ \mathbf{x}^{\mathbf{v}^*} x v ∗ a + 1 a+1 a + 1 t − ( a + 1 ) < t − a t-(a+1)<t-a t − ( a + 1 ) < t − a f ′ f' f ′ f f f
□ \square □
The last ingredient allows us to pass from L L L L ′ L' L ′
Lemma 4.8. In the setting of Setup 4.4, let L ′ = [ 0 L , b m − 1 ] L'=[0_L,b_{m-1}] L ′ = [ 0 L , b m − 1 ] A ′ = A ∩ [ m − 1 ] A'=A\cap[m-1] A ′ = A ∩ [ m − 1 ] J A ′ ′ ⊆ S L ′ J'_{A'}\subseteq S_{L'} J A ′ ′ ⊆ S L ′ Λ A ′ ′ ⊆ Z L ′ \Lambda'_{A'}\subseteq\mathbb{Z}^{L'} Λ A ′ ′ ⊆ Z L ′ ( L ′ , A ′ ) (L',A') ( L ′ , A ′ ) A ′ A' A ′ L ′ L' L ′ J A ′ ′ ⊆ J A J'_{A'}\subseteq J_A J A ′ ′ ⊆ J A u , v ∈ N L ′ \mathbf{u},\mathbf{v}\in\mathbb{N}^{L'} u , v ∈ N L ′ x u − x v ∈ I Λ A \mathbf{x}^{\mathbf{u}}-\mathbf{x}^{\mathbf{v}}\in I_{\Lambda_A} x u − x v ∈ I Λ A x u − x v ∈ I Λ A ′ ′ \mathbf{x}^{\mathbf{u}}-\mathbf{x}^{\mathbf{v}}\in I_{\Lambda'_{A'}} x u − x v ∈ I Λ A ′ ′
Proof. By (S1), the turning ranks of L ′ L' L ′ L L L A ′ A' A ′ L ′ L' L ′ L ′ L' L ′ L L L J A ′ ′ ⊆ J A J'_{A'}\subseteq J_A J A ′ ′ ⊆ J A Z L ′ \mathbb{Z}^{L'} Z L ′ Z L \mathbb{Z}^{L} Z L L ′ L' L ′ D 1 , … , D m − 1 D_1,\ldots,D_{m-1} D 1 , … , D m − 1 Λ A ′ ′ \Lambda'_{A'} Λ A ′ ′ η i \eta_i η i i ∈ [ m − 1 ] i\in[m-1] i ∈ [ m − 1 ] ρ i \rho_i ρ i i ∈ A ′ i\in A' i ∈ A ′ Λ A ′ ′ ⊆ Λ A \Lambda'_{A'}\subseteq\Lambda_A Λ A ′ ′ ⊆ Λ A x u − x v ∈ I Λ A \mathbf{x}^{\mathbf{u}}-\mathbf{x}^{\mathbf{v}}\in I_{\Lambda_A} x u − x v ∈ I Λ A u − v = ∑ i ∈ [ m ] z i η i + ∑ i ∈ A z i ′ ρ i \mathbf{u}-\mathbf{v}=\sum_{i\in[m]}z_i\eta_i+\sum_{i\in A}z'_i\rho_i u − v = ∑ i ∈ [ m ] z i η i + ∑ i ∈ A z i ′ ρ i z i , z i ′ ∈ Z z_i,z'_i\in\mathbb{Z} z i , z i ′ ∈ Z η m \eta_m η m ρ m \rho_m ρ m p p p q q q η m ( p ) = 1 \eta_m(p)=1 η m ( p ) = 1 η m ( q ) = − 1 \eta_m(q)=-1 η m ( q ) = − 1 ρ m ( p ) = 0 \rho_m(p)=0 ρ m ( p ) = 0 ρ m ( q ) = ± 1 \rho_m(q)=\pm1 ρ m ( q ) = ± 1 u − v \mathbf{u}-\mathbf{v} u − v p p p q q q z m = 0 z_m=0 z m = 0 m ∈ A m\in A m ∈ A z m ′ = 0 z'_m=0 z m ′ = 0 u − v ∈ Λ A ′ ′ \mathbf{u}-\mathbf{v}\in\Lambda_{A'}^\prime u − v ∈ Λ A ′ ′ x u − x v ∈ I Λ A ′ ′ \mathbf{x}^{\mathbf{u}}-\mathbf{x}^{\mathbf{v}}\in I_{\Lambda_{A'}^\prime} x u − x v ∈ I Λ A ′ ′
Proof of Proposition 4.1. We proceed by induction on m m m
The case m = 1 m=1 m = 1 Then L = { 0 L , c 1 , d 1 , 1 L } ≅ C 2 × C 2 L=\{0_L,c_1,d_1,1_L\}\cong C_2\times C_2 L = { 0 L , c 1 , d 1 , 1 L } ≅ C 2 × C 2 A = ∅ A=\varnothing A = ∅ J A = ( x 0 L x 1 L − x c 1 x d 1 ) J_A=(x_{0_L}x_{1_L}-x_{c_1}x_{d_1}) J A = ( x 0 L x 1 L − x c 1 x d 1 ) A = { 1 } A=\{1\} A = { 1 } S L / J A ≅ K [ x 0 L , x 1 L , x c 1 ] / ( x 0 L x 1 L − x c 1 2 ) S_L/J_A\cong K[x_{0_L},x_{1_L},x_{c_1}]/(x_{0_L}x_{1_L}-x_{c_1}^2) S L / J A ≅ K [ x 0 L , x 1 L , x c 1 ] / ( x 0 L x 1 L − x c 1 2 ) x d 1 x_{d_1} x d 1 x c 1 x_{c_1} x c 1 S L / J A S_L/J_A S L / J A x 0 L x 1 L − x c 1 x d 1 x_{0_L}x_{1_L}-x_{c_1}x_{d_1} x 0 L x 1 L − x c 1 x d 1 x 0 L x 1 L − x c 1 2 x_{0_L}x_{1_L}-x_{c_1}^2 x 0 L x 1 L − x c 1 2 x 0 L x_{0_L} x 0 L x 1 L x_{1_L} x 1 L x 0 L x_{0_L} x 0 L J A J_A J A J A = I Λ A J_A=I_{\Lambda_A} J A = I Λ A J A J_A J A I Λ A I_{\Lambda_A} I Λ A J A J_A J A K ℓ 1 K\ell_1 K ℓ 1 1 ∈ A 1\in A 1 ∈ A I Λ A I_{\Lambda_A} I Λ A ( ℓ i : i ∈ A ) (\ell_i:i\in A) ( ℓ i : i ∈ A )
The inductive step. Let m ≥ 2 m\geq2 m ≥ 2 m m m L ′ L' L ′ A ′ A' A ′ f = x u − x v ∈ I Λ A f=\mathbf{x}^{\mathbf{u}}-\mathbf{x}^{\mathbf{v}}\in I_{\Lambda_A} f = x u − x v ∈ I Λ A d = deg f d=\deg f d = deg f I Λ A I_{\Lambda_A} I Λ A d d d
We first make two reductions. Assume that m ∈ A m\in A m ∈ A B n = D m ∈ A B_n=D_m\in A B n = D m ∈ A B n B_n B n ℓ m = ± ( x γ n − 1 − x q ) \ell_m=\pm(x_{\gamma_{n-1}}-x_q) ℓ m = ± ( x γ n − 1 − x q ) g = x q − x γ n − 1 g=x_q-x_{\gamma_{n-1}} g = x q − x γ n − 1 x q x_q x q x u \mathbf{x}^{\mathbf{u}} x u x v \mathbf{x}^{\mathbf{v}} x v x γ n − 1 x_{\gamma_{n-1}} x γ n − 1 f f f x p x_p x p γ n − 1 ≠ p \gamma_{n-1}\neq p γ n − 1 = p m ∈ A m\in A m ∈ A x q x_q x q x u \mathbf{x}^{\mathbf{u}} x u x v \mathbf{x}^{\mathbf{v}} x v x u \mathbf{x}^{\mathbf{u}} x u x v \mathbf{x}^{\mathbf{v}} x v f f f d = 2 d=2 d = 2 I Λ A I_{\Lambda_A} I Λ A x u \mathbf{x}^{\mathbf{u}} x u x v \mathbf{x}^{\mathbf{v}} x v
We now distinguish two cases, according to whether x p x_p x p
Case 1: x p x_p x p x u \mathbf{x}^{\mathbf{u}} x u x v \mathbf{x}^{\mathbf{v}} x v We claim that x q x_q x q m ∈ A m\in A m ∈ A m ∉ A m\notin A m ∈ / A k 0 = n ≥ 2 k_0=n\geq2 k 0 = n ≥ 2 t = 0 t=0 t = 0 F = { p , q } F=\{p,q\} F = { p , q } δ F \delta_F δ F A A A u ( p ) + u ( q ) = v ( p ) + v ( q ) \mathbf{u}(p)+\mathbf{u}(q)=\mathbf{v}(p)+\mathbf{v}(q) u ( p ) + u ( q ) = v ( p ) + v ( q ) u ( q ) = v ( q ) \mathbf{u}(q)=\mathbf{v}(q) u ( q ) = v ( q ) u ( p ) = v ( p ) = 0 \mathbf{u}(p)=\mathbf{v}(p)=0 u ( p ) = v ( p ) = 0 x u \mathbf{x}^{\mathbf{u}} x u x v \mathbf{x}^{\mathbf{v}} x v u ( q ) = v ( q ) = 0 \mathbf{u}(q)=\mathbf{v}(q)=0 u ( q ) = v ( q ) = 0 L = L ′ ⊔ { p , q } L=L'\sqcup\{p,q\} L = L ′ ⊔ { p , q } f ∈ S L ′ f\in S_{L'} f ∈ S L ′ f ∈ I Λ A ′ f\in I_{\Lambda_{A'}} f ∈ I Λ A ′ m m m f f f ( L ′ , A ′ ) (L',A') ( L ′ , A ′ ) ( L , A ) (L,A) ( L , A ) d = 1 d=1 d = 1 f ∈ ( ℓ i : i ∈ A ′ ) ⊆ ( ℓ i : i ∈ A ) f\in(\ell_i:i\in A')\subseteq(\ell_i:i\in A) f ∈ ( ℓ i : i ∈ A ′ ) ⊆ ( ℓ i : i ∈ A ) d = 2 d=2 d = 2 f ∈ J A ′ ⊆ J A f\in J_{A'}\subseteq J_A f ∈ J A ′ ⊆ J A d ≥ 3 d\geq3 d ≥ 3 f f f I Λ A ′ I_{\Lambda_{A'}} I Λ A ′ d d d I Λ A I_{\Lambda_A} I Λ A
Case 2: x p x_p x p Replacing f f f − f -f − f x p ∣ x u x_p\mid\mathbf{x}^{\mathbf{u}} x p ∣ x u x p ∤ x v x_p\nmid\mathbf{x}^{\mathbf{v}} x p ∤ x v x v \mathbf{x}^{\mathbf{v}} x v x p x_p x p x v \mathbf{x}^{\mathbf{v}} x v x w x_w x w w ∈ F w\in F w ∈ F f f f I Λ A I_{\Lambda_A} I Λ A d d d f f f x p ∣ x u x_p\mid\mathbf{x}^{\mathbf{u}} x p ∣ x u x p ∤ x v x_p\nmid\mathbf{x}^{\mathbf{v}} x p ∤ x v
Subcase 2a: k 0 ≠ 1 k_0\neq1 k 0 = 1 r = 1 r=1 r = 1 In this case no move is needed. Indeed, δ F \delta_F δ F A A A δ F ( v ) = δ F ( u ) ≥ u ( p ) ≥ 1 \delta_F(\mathbf{v})=\delta_F(\mathbf{u})\geq\mathbf{u}(p)\geq1 δ F ( v ) = δ F ( u ) ≥ u ( p ) ≥ 1 x w ∣ x v x_w\mid\mathbf{x}^{\mathbf{v}} x w ∣ x v w ∈ F w\in F w ∈ F
Subcase 2b: k 0 = 1 k_0=1 k 0 = 1 r > 1 r>1 r > 1 This is the only point where condition (ii) of Definition 2.3 is used. Here δ F \delta_F δ F A A A E 1 = D r − 1 E_1=D_{r-1} E 1 = D r − 1 α 1 ∈ F \alpha_1\in F α 1 ∈ F E 1 E_1 E 1 F F F E 1 ∩ B ⊆ { α 0 , α 1 } E_1\cap B\subseteq\{\alpha_0,\alpha_1\} E 1 ∩ B ⊆ { α 0 , α 1 } α 0 ∉ F \alpha_0\notin F α 0 ∈ / F δ F \delta_F δ F A A A ψ \psi ψ
Step 1: construction of ψ \psi ψ By (S2), r r r B 1 = D r ∉ A B_1=D_r\notin A B 1 = D r ∈ / A B 2 ∈ A B_2\in A B 2 ∈ A k 0 = 1 k_0=1 k 0 = 1 r + 1 ∈ A r+1\in A r + 1 ∈ A r − 1 ∉ A r-1\notin A r − 1 ∈ / A E 1 ∉ A E_1\notin A E 1 ∈ / A r ′ = max ( T ( L ) ∩ { 2 , … , r − 1 } ) r'=\max\bigl(\mathcal{T}(L)\cap\{2,\ldots,r-1\}\bigr) r ′ = max ( T ( L ) ∩ { 2 , … , r − 1 } ) r ′ = 1 r'=1 r ′ = 1 h = r − r ′ ≥ 1 h=r-r'\ge1 h = r − r ′ ≥ 1 k k k r ′ < k < r r'<k<r r ′ < k < r [ a r ′ , b r ′ ] = D r ′ ∪ ⋯ ∪ D r [a_{r'},b_{r'}]=D_{r'}\cup\cdots\cup D_r [ a r ′ , b r ′ ] = D r ′ ∪ ⋯ ∪ D r β y \beta_y β y β y ′ \beta'_y β y ′ − h ≤ y ≤ 1 -h\le y\le1 − h ≤ y ≤ 1 λ y + h \lambda_{y+h} λ y + h μ y + h \mu_{y+h} μ y + h B ′ = [ a r ′ , b r ] = { β y , β y ′ : − h ≤ y ≤ 1 } ≅ C h + 2 × C 2 B'=[a_{r'},b_r]=\{\beta_y,\beta'_y:-h\le y\le1\}\cong C_{h+2}\times C_2 B ′ = [ a r ′ , b r ] = { β y , β y ′ : − h ≤ y ≤ 1 } ≅ C h + 2 × C 2 E i = D r − i E_i=D_{r-i} E i = D r − i 0 ≤ i ≤ h 0\le i\le h 0 ≤ i ≤ h
E i = [ β − i , β 1 − i ′ ] = { β − i , β − i ′ , β 1 − i , β 1 − i ′ } , { c r − i , d r − i } = { β − i ′ , β 1 − i } . E_i=[\beta_{-i},\beta'_{1-i}]=\{\beta_{-i},\beta'_{-i},\beta_{1-i},\beta'_{1-i}\},\qquad\{c_{r-i},d_{r-i}\}=\{\beta'_{-i},\beta_{1-i}\}. E i = [ β − i , β 1 − i ′ ] = { β − i , β − i ′ , β 1 − i , β 1 − i ′ } , { c r − i , d r − i } = { β − i ′ , β 1 − i } . In particular, E 0 = B 1 E_0=B_1 E 0 = B 1 β 0 = a r = α 0 \beta_0=a_r=\alpha_0 β 0 = a r = α 0 β 0 ′ = b r − 1 = α 1 \beta'_0=b_{r-1}=\alpha_1 β 0 ′ = b r − 1 = α 1 β 1 ′ = b r = γ 1 \beta'_1=b_r=\gamma_1 β 1 ′ = b r = γ 1 { β 0 ′ , β 1 ′ } = { c r , d r } = { γ 0 , α 1 } \{\beta'_0,\beta'_1\}=\{c_r,d_r\}=\{\gamma_0,\alpha_1\} { β 0 ′ , β 1 ′ } = { c r , d r } = { γ 0 , α 1 } β 1 ′ = γ 0 \beta'_1=\gamma_0 β 1 ′ = γ 0 B ∩ B ′ = { α 0 , α 1 , γ 0 , γ 1 } B\cap B'=\{\alpha_0,\alpha_1,\gamma_0,\gamma_1\} B ∩ B ′ = { α 0 , α 1 , γ 0 , γ 1 } B ′ B' B ′ B ′ B' B ′ r + 1 r+1 r + 1 B ′ B' B ′ r + 1 r+1 r + 1 b r = γ 1 b_r=\gamma_1 b r = γ 1 B B B r + 1 r+1 r + 1 α 0 , γ 0 , α 1 , γ 1 , α 2 \alpha_0,\gamma_0,\alpha_1,\gamma_1,\alpha_2 α 0 , γ 0 , α 1 , γ 1 , α 2 r ′ > 1 r'>1 r ′ > 1 r ′ ∈ T ( L ) r'\in T(L) r ′ ∈ T ( L ) E h = D r ′ ∉ A E_h=D_{r'}\notin\mathcal{A} E h = D r ′ ∈ / A b r ′ − 1 = a r ′ + 1 = β 1 − h ′ b_{r'-1}=a_{r'+1}=\beta'_{1-h} b r ′ − 1 = a r ′ + 1 = β 1 − h ′ D i D_i D i i < r ′ i<r' i < r ′ B ∪ B ′ B\cup B' B ∪ B ′ β − h = a r ′ \beta_{-h}=a_{r'} β − h = a r ′ β 1 − h \beta_{1-h} β 1 − h D i D_i D i i + 1 ≤ r ′ i+1\le r' i + 1 ≤ r ′ B ′ B' B ′ r ′ r' r ′ β − h \beta_{-h} β − h r ′ − 1 r'-1 r ′ − 1 β − h ′ , β 1 − h \beta'_{-h},\beta_{1-h} β − h ′ , β 1 − h r ′ r' r ′ B B B r − 1 ≥ r ′ r-1\ge r' r − 1 ≥ r ′ r ′ r' r ′ α 0 = β 0 \alpha_0=\beta_0 α 0 = β 0 h = 1 h=1 h = 1 β 0 = β 1 − h \beta_0=\beta_{1-h} β 0 = β 1 − h β − h ′ ∈ D i \beta'_{-h}\in D_i β − h ′ ∈ D i rank L ( β − h ′ ) = r ′ \operatorname{rank}_L(\beta'_{-h})=r' rank L ( β − h ′ ) = r ′ i = r ′ − 1 i=r'-1 i = r ′ − 1 β − h ′ = b r ′ − 1 = β 1 − h \beta'_{-h}=b_{r'-1}=\beta_{1-h} β − h ′ = b r ′ − 1 = β 1 − h
Define ζ ( − h ) = 0 \zeta(-h)=0 ζ ( − h ) = 0 − h < y ≤ 1 -h<y\le1 − h < y ≤ 1 ζ ( y ) = ζ ( y − 1 ) + 1 \zeta(y)=\zeta(y-1)+1 ζ ( y ) = ζ ( y − 1 ) + 1 E 1 − y ∈ A E_{1-y}\in\mathcal{A} E 1 − y ∈ A ζ ( y ) = 0 \zeta(y)=0 ζ ( y ) = 0 E 0 , E 1 ∈ A c E_0,E_1\in\mathcal{A}^c E 0 , E 1 ∈ A c ζ ( 0 ) = ζ ( 1 ) = 0 \zeta(0)=\zeta(1)=0 ζ ( 0 ) = ζ ( 1 ) = 0 r ′ > 1 r'>1 r ′ > 1 ζ ( 1 − h ) = 0 \zeta(1-h)=0 ζ ( 1 − h ) = 0 E h ∉ A E_h\notin\mathcal{A} E h ∈ / A ψ : L → Z \psi:L\to\mathbb{Z} ψ : L → Z ψ ( β y ) = ζ ( y ) \psi(\beta_y)=\zeta(y) ψ ( β y ) = ζ ( y ) ψ ( β y ′ ) = ζ ( y ) + 1 \psi(\beta'_y)=\zeta(y)+1 ψ ( β y ′ ) = ζ ( y ) + 1 − h ≤ y ≤ 1 -h\le y\le1 − h ≤ y ≤ 1 ψ ( α j ) = ψ ( γ j ) = 1 \psi(\alpha_j)=\psi(\gamma_j)=1 ψ ( α j ) = ψ ( γ j ) = 1 2 ≤ j ≤ n 2\le j\le n 2 ≤ j ≤ n ψ ( w ) = 0 \psi(w)=0 ψ ( w ) = 0 α j , γ j ∉ B ′ \alpha_j,\gamma_j\notin B' α j , γ j ∈ / B ′ j ≥ 2 j\ge2 j ≥ 2 ψ ≥ 0 \psi\ge0 ψ ≥ 0 ψ ( p ) = 1 \psi(p)=1 ψ ( p ) = 1 ψ \psi ψ δ F \delta_F δ F B B B F = { α j , γ j : 1 ≤ j ≤ n } F=\{\alpha_j,\gamma_j:1\le j\le n\} F = { α j , γ j : 1 ≤ j ≤ n } ψ ( α 0 ) = ψ ( β 0 ) = 0 \psi(\alpha_0)=\psi(\beta_0)=0 ψ ( α 0 ) = ψ ( β 0 ) = 0 ψ ( γ 0 ) = ψ ( β 1 ) = 0 \psi(\gamma_0)=\psi(\beta_1)=0 ψ ( γ 0 ) = ψ ( β 1 ) = 0 ψ ( α 1 ) = ψ ( β 0 ′ ) = 1 \psi(\alpha_1)=\psi(\beta'_0)=1 ψ ( α 1 ) = ψ ( β 0 ′ ) = 1 ψ ( γ 1 ) = ψ ( β 1 ′ ) = 1 \psi(\gamma_1)=\psi(\beta'_1)=1 ψ ( γ 1 ) = ψ ( β 1 ′ ) = 1
We show that ψ \psi ψ A \mathcal{A} A B 1 , … , B n B_1,\ldots,B_n B 1 , … , B n ψ \psi ψ δ F \delta_F δ F B B B k 0 k_0 k 0 E 1 − y E_{1-y} E 1 − y − h < y ≤ 0 -h<y\le0 − h < y ≤ 0 β y − 1 \beta_{y-1} β y − 1 β y ′ \beta'_y β y ′ ψ ( β y − 1 ) + ψ ( β y ′ ) \psi(\beta_{y-1})+\psi(\beta'_y) ψ ( β y − 1 ) + ψ ( β y ′ ) ψ ( β y − 1 ′ ) + ψ ( β y ) \psi(\beta'_{y-1})+\psi(\beta_y) ψ ( β y − 1 ′ ) + ψ ( β y ) ζ ( y − 1 ) + ζ ( y ) + 1 \zeta(y-1)+\zeta(y)+1 ζ ( y − 1 ) + ζ ( y ) + 1 E 1 − y ∈ A E_{1-y}\in\mathcal{A} E 1 − y ∈ A ψ ( β y ) = ζ ( y ) = ζ ( y − 1 ) + 1 = ψ ( β y − 1 ′ ) \psi(\beta_y)=\zeta(y)=\zeta(y-1)+1=\psi(\beta'_{y-1}) ψ ( β y ) = ζ ( y ) = ζ ( y − 1 ) + 1 = ψ ( β y − 1 ′ ) r ′ > 1 r'>1 r ′ > 1 D i D_i D i i < r ′ i<r' i < r ′ B ∪ B ′ B\cup B' B ∪ B ′ β − h \beta_{-h} β − h β 1 − h \beta_{1-h} β 1 − h ψ \psi ψ ζ ( − h ) = ζ ( 1 − h ) = 0 ; h e n c e \zeta(-h)=\zeta(1-h)=0; hence ζ ( − h ) = ζ ( 1 − h ) = 0 ; h e n ce D i D_i D i ψ \psi ψ A \mathcal{A} A
Step 2: moving a variable indexed by F F F x v x^{\mathbf v} x v If x v x^{\mathbf v} x v x w x_w x w w ∈ F w\in F w ∈ F k 0 = 1 k_0=1 k 0 = 1 γ 0 \gamma_0 γ 0 F F F ω \omega ω ω ( v ) = ω ( u ) ≥ ω ( p ) ≥ 1 \omega(\mathbf v)=\omega(\mathbf u)\ge\omega(p)\ge1 ω ( v ) = ω ( u ) ≥ ω ( p ) ≥ 1 x γ 0 ∣ x v x_{\gamma_0}\mid x^{\mathbf v} x γ 0 ∣ x v ψ ( v ) = ψ ( u ) ≥ ψ ( p ) = 1 \psi(\mathbf v)=\psi(\mathbf u)\ge\psi(p)=1 ψ ( v ) = ψ ( u ) ≥ ψ ( p ) = 1 x v x^{\mathbf v} x v x w x_w x w w ∉ F w\notin F w ∈ / F ψ ( w ) ≥ 1 \psi(w)\ge1 ψ ( w ) ≥ 1 ψ \psi ψ w = β y ′ w=\beta'_y w = β y ′ − h ≤ y ≤ − 1 -h\le y\le-1 − h ≤ y ≤ − 1 w = β y w=\beta_y w = β y − h < y ≤ − 1 -h<y\le-1 − h < y ≤ − 1 ζ ( y ) ≥ 1 \zeta(y)\ge1 ζ ( y ) ≥ 1 E 1 − y ∈ A E_{1-y}\in\mathcal{A} E 1 − y ∈ A ± ( x β y − x β y − 1 ′ ) \pm(x_{\beta_y}-x_{\beta'_{y-1}}) ± ( x β y − x β y − 1 ′ ) x β y x_{\beta_y} x β y x β y − 1 ′ x_{\beta'_{y-1}} x β y − 1 ′ x v x^{\mathbf v} x v x p x_p x p x γ 0 x_{\gamma_0} x γ 0 w = β y ′ w=\beta'_y w = β y ′ − h ≤ y ≤ − 1 -h\le y\le-1 − h ≤ y ≤ − 1 γ 0 = β 1 \gamma_0=\beta_1 γ 0 = β 1 β y ′ \beta'_y β y ′ β y ′ ∧ β 1 = β y \beta'_y\wedge\beta_1=\beta_y β y ′ ∧ β 1 = β y β y ′ ∨ β 1 = β 1 ′ = γ 1 \beta'_y\vee\beta_1=\beta'_1=\gamma_1 β y ′ ∨ β 1 = β 1 ′ = γ 1 x γ 0 x β y ′ x_{\gamma_0}x_{\beta'_y} x γ 0 x β y ′ x β y x γ 1 x_{\beta_y}x_{\gamma_1} x β y x γ 1 x v x^{\mathbf v} x v x γ 1 x_{\gamma_1} x γ 1 γ 1 ∈ F \gamma_1\in F γ 1 ∈ F x p x_p x p γ 1 ≠ p \gamma_1\ne p γ 1 = p n ≥ 2 n\ge2 n ≥ 2 β y ≠ p \beta_y\ne p β y = p f f f
In all cases f f f
Theorem 4.9. Let L L L m + 1 m+1 m + 1 A ⊆ [ m ] A \subseteq[m] A ⊆ [ m ] J A = I Λ A J_A=I_{\Lambda_A} J A = I Λ A J A J_A J A dim ( S L / J A ) = m + 2 − ∣ A ∣ \dim(S_L/J_A)=m+2-|A| dim ( S L / J A ) = m + 2 − ∣ A ∣
Proof. We have J A ⊆ I Λ A J_A\subseteq I_{\Lambda_A} J A ⊆ I Λ A d ≥ 1 d\geq1 d ≥ 1 I Λ A I_{\Lambda_A} I Λ A d d d J A J_A J A d ≤ 2 d\leq2 d ≤ 2 d ≥ 3 d\geq3 d ≥ 3 I Λ A I_{\Lambda_A} I Λ A d d d J A J_A J A I Λ A I_{\Lambda_A} I Λ A I Λ A ⊆ J A I_{\Lambda_A}\subseteq J_A I Λ A ⊆ J A J A = I Λ A J_A=I_{\Lambda_A} J A = I Λ A Λ A \Lambda_A Λ A I Λ A I_{\Lambda_A} I Λ A dim ( S L / J A ) = ∣ L ∣ − rank Z Λ A = ( 2 m + 2 ) − ( m + ∣ A ∣ ) \dim(S_L/J_A)=|L|-\operatorname{rank}_{\mathbb Z}\Lambda_A=(2m+2)-(m+|A|) dim ( S L / J A ) = ∣ L ∣ − rank Z Λ A = ( 2 m + 2 ) − ( m + ∣ A ∣ ) [8 ] for the dimension formula for lattice ideals. □
Proof of Theorem 2.4. If J A J_A J A A A A A A A J A J_A J A
We conclude this section with some observations on S L / J A S_L/J_A S L / J A A ⊆ [ m ] A\subseteq[m] A ⊆ [ m ]
Remark 4.10. Let A ⊆ [ m ] A\subseteq[m] A ⊆ [ m ] [9 ] , K [ L ] K[L] K [ L ] L L L j ∈ [ m ] j\in[m] j ∈ [ m ] ℓ j = x c j − x d j \ell_j=x_{c_j}-x_{d_j} ℓ j = x c j − x d j L L L j j j x 0 L x_{0_L} x 0 L x 1 L x_{1_L} x 1 L m + 1 m+1 m + 1 [4 Theorem 4.1] , the residue classes of x 0 L , ℓ 1 , … , ℓ m , x 1 L x_{0_L},\ell_1,\ldots,\ell_m,x_{1_L} x 0 L , ℓ 1 , … , ℓ m , x 1 L K [ L ] K[L] K [ L ] ℓ i \ell_i ℓ i i ∈ A i\in A i ∈ A [2 Exercise 1.5.23] . It follows that:
S L / J A = K [ L ] / ( ℓ i : i ∈ A ) S_L/J_A=K[L]/(\ell_i:i\in A) S L / J A = K [ L ] / ( ℓ i : i ∈ A ) [2 Theorem 2.1.3(a)] ;
the reduced Hilbert–Poincaré series of S L / J A S_L/J_A S L / J A
W L ( t ) ( 1 − t ) m + 2 − ∣ A ∣ , \frac{W_L(t)}{(1-t)^{m+2-|A|}}, ( 1 − t ) m + 2 − ∣ A ∣ W L ( t ) , where W L ( t ) W_L(t) W L ( t ) descent polynomial of P L P_L P L [11 Chapter 3] ), which represents the h h h K [ L ] K[L] K [ L ]
the regularity of S L / J A S_L/J_A S L / J A m − rank ( P L ) m-\operatorname{rank}(P_L) m − rank ( P L ) K [ L ] K[L] K [ L ] [5 Theorem 1.1] ).
Since J A J_A J A ℓ i \ell_i ℓ i i ∈ A i\in A i ∈ A S L / J A S_L/J_A S L / J A T T T K K K x w x_w x w w ∈ L ∖ { d i : i ∈ A } w\in L\setminus\{d_i:i\in A\} w ∈ L ∖ { d i : i ∈ A } σ A : S L → T \sigma_A:S_L\to T σ A : S L → T K K K σ A ( x d i ) = x c i \sigma_A(x_{d_i})=x_{c_i} σ A ( x d i ) = x c i i ∈ A i\in A i ∈ A σ A ( x w ) = x w \sigma_A(x_w)=x_w σ A ( x w ) = x w ( ℓ i : i ∈ A ) (\ell_i:i\in A) ( ℓ i : i ∈ A ) J ‾ A = σ A ( I L ) \overline{J}_A=\sigma_A(I_L) J A = σ A ( I L ) S L / J A ≅ T / J ‾ A S_L/J_A\cong T/\overline{J}_A S L / J A ≅ T / J A K K K
Corollary 4.11. Let L L L m + 1 m+1 m + 1 A ⊆ [ m ] A\subseteq[m] A ⊆ [ m ]
(i) J ‾ A \overline{J}_A J A
(ii) rank ( P L ) = m − 1 \operatorname{rank}(P_L)=m-1 rank ( P L ) = m − 1
(iii) L ≅ C m + 1 × C 2 L\cong C_{m+1}\times C_2 L ≅ C m + 1 × C 2
(iv) T ( L ) = ∅ T(L)=\varnothing T ( L ) = ∅
Consequently, J A J_A J A S L / J A S_L/J_A S L / J A J ‾ A \overline{J}_A J A L ≅ C m + 1 × C 2 L\cong C_{m+1}\times C_2 L ≅ C m + 1 × C 2 A ⊆ [ m ] A\subseteq[m] A ⊆ [ m ]
Proof. Since L L L ∣ P L ∣ = m + 1 |P_L|=m+1 ∣ P L ∣ = m + 1 rank ( P L ) ≤ m − 1 \operatorname{rank}(P_L)\leq m-1 rank ( P L ) ≤ m − 1 reg ( T / J ‾ A ) = reg ( S L / J A ) = m − rank ( P L ) ≥ 1 \operatorname{reg}(T/\overline{J}_A)=\operatorname{reg}(S_L/J_A)=m-\operatorname{rank}(P_L)\geq1 reg ( T / J A ) = reg ( S L / J A ) = m − rank ( P L ) ≥ 1 J ‾ A \overline{J}_A J A reg ( T / J ‾ A ) = 1 \operatorname{reg}(T/\overline{J}_A)=1 reg ( T / J A ) = 1 ⇔ \Leftrightarrow ⇔ rank ( P L ) = m − 1 \operatorname{rank}(P_L)=m-1 rank ( P L ) = m − 1 P L = C ∪ { z } P_L=C\cup\{z\} P L = C ∪ { z } C = { p 1 < ⋯ < p m } C=\{p_1<\cdots<p_m\} C = { p 1 < ⋯ < p m } z z z C C C { p 1 } \{p_1\} { p 1 } P L ∖ { p m } P_L \setminus\{p_m\} P L ∖ { p m } P L P_L P L L L L 0 L 0_L 0 L 1 L 1_L 1 L L ≅ J ( C ) × J ( { z } ) ≅ C m + 1 × C 2 L \cong\mathcal{J}(C) \times\mathcal{J}(\{z\}) \cong C_{m+1} \times C_2 L ≅ J ( C ) × J ({ z }) ≅ C m + 1 × C 2 ⇔ \Leftrightarrow ⇔ T ( L ) = ∅ T(L) = \varnothing T ( L ) = ∅ L L L ⇔ \Leftrightarrow ⇔ T ( L ) = ∅ T(L) = \varnothing T ( L ) = ∅ A ⊆ [ m ] A \subseteq[m] A ⊆ [ m ] Theorem 2.4 and Remark 4.10. □
A toric parametrization
In this section we show that, when A A A S L / J A S_L/J_A S L / J A K [ L ] K[L] K [ L ] [9 ] .
Let L L L m + 1 m+1 m + 1 R L = K [ z , y p : p ∈ P L ] \mathcal{R}_L = K[z,y_p : p \in P_L] R L = K [ z , y p : p ∈ P L ] K K K
φ : S L ⟶ R L , φ ( x α ) = z ∏ p ∈ i α y p . \varphi:S_L \longrightarrow\mathcal{R}_L,\qquad\varphi(x_\alpha)=z\prod_{p\in i_\alpha}y_p. φ : S L ⟶ R L , φ ( x α ) = z p ∈ i α ∏ y p . By [9 ] , ker φ = I L \ker\varphi=I_L ker φ = I L K [ L ] K[L] K [ L ] K [ z ∏ p ∈ i α y p : α ∈ L ] K[z\prod_{p\in i_\alpha}y_p:\alpha\in L] K [ z ∏ p ∈ i α y p : α ∈ L ] φ \varphi φ
β : Z L ⟶ Γ L = Z e z ⊕ ⨁ p ∈ P L Z e p , β ( e α ) = e z + ∑ p ∈ i α e p . \beta:\mathbb{Z}^{L}\longrightarrow\Gamma_L=\mathbb{Z}e_z\oplus\bigoplus_{p\in P_L}\mathbb{Z}e_p,\qquad\beta(e_\alpha)=e_z+\sum_{p\in i_\alpha}e_p. β : Z L ⟶ Γ L = Z e z ⊕ p ∈ P L ⨁ Z e p , β ( e α ) = e z + p ∈ i α ∑ e p . Recall that, if ϕ \phi ϕ K K K S L S_L S L ker ϕ \ker\phi ker ϕ [8 Chapter 3] . In particular, I L = ker φ = I ker Z β I_L=\ker\varphi=I_{\ker_{\mathbb{Z}}\beta} I L = ker φ = I k e r Z β
For every i ∈ [ m ] i\in[m] i ∈ [ m ] i a i = i c i ∩ i d i i_{a_i}=i_{c_i}\cap i_{d_i} i a i = i c i ∩ i d i ∣ i c i ∣ = ∣ i d i ∣ = ∣ i a i ∣ + 1 |i_{c_i}|=|i_{d_i}|=|i_{a_i}|+1 ∣ i c i ∣ = ∣ i d i ∣ = ∣ i a i ∣ + 1 p i , p i ′ ∈ P L p_i,p_i'\in P_L p i , p i ′ ∈ P L
i c i = i a i ⊔ { p i } , i d i = i a i ⊔ { p i ′ } . i_{c_i}=i_{a_i}\sqcup\{p_i\},\qquad i_{d_i}=i_{a_i}\sqcup\{p_i'\}. i c i = i a i ⊔ { p i } , i d i = i a i ⊔ { p i ′ } . and p i ≠ p i ′ p_i\ne p_i' p i = p i ′ c i ≠ d i c_i\ne d_i c i = d i p i p_i p i L L L c i c_i c i d i d_i d i p i ′ p_i' p i ′
β ( ρ i ) = β ( e c i ) − β ( e d i ) = e p i − e p i ′ . \beta(\rho_i)=\beta(e_{c_i})-\beta(e_{d_i})=e_{p_i}-e_{p_i'}. β ( ρ i ) = β ( e c i ) − β ( e d i ) = e p i − e p i ′ . For A ⊆ [ m ] A\subseteq[m] A ⊆ [ m ] ∼ A \sim_A ∼ A P L P_L P L p i ∼ A p i ′ p_i\sim_A p_i' p i ∼ A p i ′ i ∈ A i\in A i ∈ A [ p ] [p] [ p ] p ∈ P L p\in P_L p ∈ P L R L , A = K [ z , y [ p ] : [ p ] ∈ P L / ∼ A ] \mathcal{R}_{L,A}=K[z,y_{[p]}:[p]\in P_L/\sim_A] R L , A = K [ z , y [ p ] : [ p ] ∈ P L / ∼ A ] K K K
φ A : S L ⟶ R L , A , φ A ( x α ) = z ∏ p ∈ i α y [ p ] . \varphi_A:S_L\longrightarrow\mathcal{R}_{L,A},\qquad\varphi_A(x_\alpha)=z\prod_{p\in i_\alpha}y_{[p]}. φ A : S L ⟶ R L , A , φ A ( x α ) = z p ∈ i α ∏ y [ p ] . Thus φ A \varphi_A φ A φ \varphi φ y p i y_{p_i} y p i y p i ′ y_{p_i'} y p i ′ i ∈ A i\in A i ∈ A φ ∅ = φ \varphi_\varnothing=\varphi φ ∅ = φ T A = K [ φ A ( x α ) : α ∈ L ] ⊆ R L , A T_A=K[\varphi_A(x_\alpha):\alpha\in L]\subseteq\mathcal{R}_{L,A} T A = K [ φ A ( x α ) : α ∈ L ] ⊆ R L , A
Lemma 5.1. The homomorphism β \beta β ker Z β = Λ ∅ \ker_{\mathbb{Z}}\beta=\Lambda_\varnothing ker Z β = Λ ∅
Proof. Since i 0 L = ∅ i_{0_L}=\varnothing i 0 L = ∅ β ( e 0 L ) = e z \beta(e_{0_L})=e_z β ( e 0 L ) = e z p ∈ P L p\in P_L p ∈ P L U = { p ′ ∈ P L : p ′ ≤ p } U=\{p'\in P_L:p'\leq p\} U = { p ′ ∈ P L : p ′ ≤ p } U ∖ { p } U\setminus\{p\} U ∖ { p } P L P_L P L α , α ′ ∈ L \alpha,\alpha'\in L α , α ′ ∈ L i α = U i_\alpha=U i α = U i α ′ = U ∖ { p } i_{\alpha'}=U\setminus\{p\} i α ′ = U ∖ { p } β ( e α ) − β ( e α ′ ) = e p \beta(e_\alpha)-\beta(e_{\alpha'})=e_p β ( e α ) − β ( e α ′ ) = e p β \beta β
By [9 ] , I L = J ∅ I_L=J_\varnothing I L = J ∅ Theorem 3.7 gives I L = I Λ ∅ I_L=I_{\Lambda_\varnothing} I L = I Λ ∅ I ker Z β = I Λ ∅ I_{\ker_{\mathbb{Z}}\beta}=I_{\Lambda_\varnothing} I k e r Z β = I Λ ∅ z ∈ Λ \mathbf{z}\in\Lambda z ∈ Λ x z + − x z − ∈ I Λ x^{z^+}-x^{z^-}\in I_\Lambda x z + − x z − ∈ I Λ Λ ⊆ Z L \Lambda\subseteq\mathbb{Z}^L Λ ⊆ Z L ker Z β = Λ ∅ \ker_{\mathbb{Z}}\beta=\Lambda_\varnothing ker Z β = Λ ∅
Proposition 5.2. Let L L L m + 1 m+1 m + 1 A ⊆ [ m ] A\subseteq[m] A ⊆ [ m ] J A ⊆ ker φ A = I Λ A J_A\subseteq\ker\varphi_A=I_{\Lambda_A} J A ⊆ ker φ A = I Λ A
In particular, T A ≅ S L / I Λ A T_A\cong S_L/I_{\Lambda_A} T A ≅ S L / I Λ A dim T A = m + 2 − ∣ A ∣ \dim T_A=m+2-|A| dim T A = m + 2 − ∣ A ∣
Proof. Let π A : Γ L → Γ L , A = Z e z ⊕ ⨁ [ p ] ∈ P L / ∼ A Z e [ p ] \pi_A:\Gamma_L\to\Gamma_{L,A}=\mathbb{Z}e_z\oplus\bigoplus_{[p]\in P_L/{\sim_A}}\mathbb{Z}e_{[p]} π A : Γ L → Γ L , A = Z e z ⊕ ⨁ [ p ] ∈ P L / ∼ A Z e [ p ] π A ( e z ) = e z \pi_A(e_z)=e_z π A ( e z ) = e z π A ( e p ) = e [ p ] \pi_A(e_p)=e_{[p]} π A ( e p ) = e [ p ] φ A \varphi_A φ A π A ∘ β \pi_A\circ\beta π A ∘ β ker φ A = I ker Z ( π A ∘ β ) \ker\varphi_A=I_{\ker_{\mathbb{Z}}(\pi_A\circ\beta)} ker φ A = I k e r Z ( π A ∘ β ) ker Z ( π A ∘ β ) = Λ A \ker_{\mathbb{Z}}(\pi_A\circ\beta)=\Lambda_A ker Z ( π A ∘ β ) = Λ A
We claim that ker Z π A \ker_{\mathbb{Z}}\pi_A ker Z π A e p i − e p i ′ e_{p_i}-e_{p_i'} e p i − e p i ′ i ∈ A i\in A i ∈ A ker Z π A \ker_{\mathbb{Z}}\pi_A ker Z π A [ p i ] = [ p i ′ ] [p_i]=[p_i'] [ p i ] = [ p i ′ ] z = λ e z + ∑ p ∈ P L λ p e p ∈ ker Z π A z=\lambda e_z+\sum_{p\in P_L}\lambda_p e_p\in\ker_{\mathbb{Z}}\pi_A z = λ e z + ∑ p ∈ P L λ p e p ∈ ker Z π A λ = 0 \lambda=0 λ = 0 ∑ p ∈ C λ p = 0 \sum_{p\in C}\lambda_p=0 ∑ p ∈ C λ p = 0 C C C ∼ A \sim_A ∼ A z z z e p − e p ′ ′ e_p-e_{p''} e p − e p ′′ p ∼ A p ′ ′ p\sim_A p'' p ∼ A p ′′ p ∼ A p ′ ′ p\sim_A p'' p ∼ A p ′′ p = q 0 , q 1 , … , q l = p ′ ′ p=q_0,q_1,\ldots,q_l=p'' p = q 0 , q 1 , … , q l = p ′′ { q j − 1 , q j } = { p i , p i ′ } \{q_{j-1},q_j\}=\{p_i,p_i'\} { q j − 1 , q j } = { p i , p i ′ } i ∈ A i\in A i ∈ A j ∈ [ l ] j\in[l] j ∈ [ l ] e p − e p ′ ′ = ∑ j = 1 l ( e q j − 1 − e q j ) e_p-e_{p''}=\sum_{j=1}^l(e_{q_{j-1}}-e_{q_j}) e p − e p ′′ = ∑ j = 1 l ( e q j − 1 − e q j ) e p i − e p i ′ e_{p_i}-e_{p_i'} e p i − e p i ′ i ∈ A i\in A i ∈ A
Let P A P_A P A Z L \mathbb{Z}^L Z L ρ i \rho_i ρ i i ∈ A i\in A i ∈ A Λ A = Λ ∅ + P A \Lambda_A=\Lambda_{\varnothing}+P_A Λ A = Λ ∅ + P A β ( ρ i ) = e p i − e p i ′ \beta(\rho_i)=e_{p_i}-e_{p_i'} β ( ρ i ) = e p i − e p i ′ ker Z π A = β ( P A ) \ker_{\mathbb{Z}}\pi_A=\beta(P_A) ker Z π A = β ( P A )
ker Z ( π A ∘ β ) = β − 1 ( β ( P A ) ) = P A + ker Z β = P A + Λ ∅ = Λ A . \ker_{\mathbb{Z}}(\pi_A\circ\beta)=\beta^{-1}(\beta(P_A))=P_A+\ker_{\mathbb{Z}}\beta=P_A+\Lambda_{\varnothing}=\Lambda_A. ker Z ( π A ∘ β ) = β − 1 ( β ( P A )) = P A + ker Z β = P A + Λ ∅ = Λ A . Here, the second equality holds because β ( z ) ∈ β ( P A ) \beta(z)\in\beta(P_A) β ( z ) ∈ β ( P A ) z − r ∈ ker Z β z-r\in\ker_{\mathbb{Z}}\beta z − r ∈ ker Z β r ∈ P A r\in P_A r ∈ P A ker φ A = I Λ A \ker\varphi_A=I_{\Lambda_A} ker φ A = I Λ A J A ⊆ I Λ A J_A\subseteq I_{\Lambda_A} J A ⊆ I Λ A T A ≅ S L / ker φ A = S L / I Λ A T_A\cong S_L/\ker\varphi_A=S_L/I_{\Lambda_A} T A ≅ S L / ker φ A = S L / I Λ A dim T A = ∣ L ∣ − rank Z Λ A = ( 2 m + 2 ) − ( m + ∣ A ∣ ) = m + 2 − ∣ A ∣ \dim T_A=|L|-\operatorname{rank}_{\mathbb{Z}}\Lambda_A=(2m+2)-(m+|A|)=m+2-|A| dim T A = ∣ L ∣ − rank Z Λ A = ( 2 m + 2 ) − ( m + ∣ A ∣ ) = m + 2 − ∣ A ∣
Corollary 5.3. Let L L L m + 1 m+1 m + 1 A ⊆ [ m ] A\subseteq[m] A ⊆ [ m ] ker φ A = J A \ker\varphi_A=J_A ker φ A = J A A A A S L / J A ≅ T A S_L/J_A\cong T_A S L / J A ≅ T A S L / J A S_L/J_A S L / J A K [ L ] K[L] K [ L ] y p i y_{p_i} y p i y p i ′ y_{p_i'} y p i ′ i ∈ A i\in A i ∈ A
Proof. If A A A J A = I Λ A = ker φ A J_A=I_{\Lambda_A}=\ker\varphi_A J A = I Λ A = ker φ A ker φ A = J A \ker\varphi_A=J_A ker φ A = J A J A J_A J A T A T_A T A A A A
The parametrization φ \varphi φ K = R K=\mathbb{R} K = R P L P_L P L P L P_L P L p p p p ′ < p p'<p p ′ < p conjunctive Bayesian network on P L P_L P L [1 ] . The possible states are the sets of events which have occurred, namely the poset ideals i α i_\alpha i α α ∈ L \alpha\in L α ∈ L θ p ∈ ( 0 , 1 ) \theta_p\in(0,1) θ p ∈ ( 0 , 1 ) p ∈ P L p\in P_L p ∈ P L θ p \theta_p θ p p p p p ′ < p p'<p p ′ < p α ∈ L \alpha\in L α ∈ L q α q_\alpha q α i α i_\alpha i α q α = ∏ p ∈ i α θ p q_\alpha=\prod_{p\in i_\alpha}\theta_p q α = ∏ p ∈ i α θ p [1 Theorem 10] , that is, q α q_\alpha q α φ ( x α ) \varphi(x_\alpha) φ ( x α ) z = 1 z=1 z = 1 y p = θ p y_p=\theta_p y p = θ p L L L q α q_\alpha q α q 0 L = 1 q_{0_L}=1 q 0 L = 1 q α q β = q α ∧ β q α ∨ β q_\alpha q_\beta=q_{\alpha\wedge\beta}q_{\alpha\vee\beta} q α q β = q α ∧ β q α ∨ β [1 Corollary 11] . This is the case A = ∅ A=\varnothing A = ∅
Our results describe what happens when some conditional probabilities are required to coincide. Since i r i = i a i ⊔ { p i } i_{r_i}=i_{a_i}\sqcup\{p_i\} i r i = i a i ⊔ { p i } i d i = i a i ⊔ { p i ′ } i_{d_i}=i_{a_i}\sqcup\{p_i'\} i d i = i a i ⊔ { p i ′ } q c i = q d i ( ∏ p ∈ i a i θ p ) ( θ p i − θ p i ′ ) q_{c_i}=q_{d_i}\left(\prod_{p\in i_{a_i}}\theta_p\right)(\theta_{p_i}-\theta_{p_i'}) q c i = q d i ( ∏ p ∈ i a i θ p ) ( θ p i − θ p i ′ ) q c i = q d i q_{c_i}=q_{d_i} q c i = q d i θ p i = θ p i ′ \theta_{p_i}=\theta_{p_i'} θ p i = θ p i ′ x c i = x d i x_{c_i}=x_{d_i} x c i = x d i p i p_i p i p i ′ p_i' p i ′ i i i A ⊆ [ m ] A\subseteq[m] A ⊆ [ m ] i ∈ A i\in A i ∈ A θ p \theta_p θ p [ p ] [p] [ p ] p p p ∼ A \sim_A ∼ A q α q_\alpha q α φ A ( x α ) \varphi_A(x_\alpha) φ A ( x α ) z = 1 z=1 z = 1 y [ p ] = θ [ p ] y_{[p]}=\theta_{[p]} y [ p ] = θ [ p ] f ∈ S L f\in S_L f ∈ S L r r r φ A ( f ) = z r ⋅ φ A ( f ) ∣ z = 1 \varphi_A(f)=z^r\cdot\varphi_A(f)|_{z=1} φ A ( f ) = z r ⋅ φ A ( f ) ∣ z = 1 f f f ( q α ) α ∈ L (q_\alpha)_{\alpha\in L} ( q α ) α ∈ L f ∈ ker φ A f\in\ker\varphi_A f ∈ ker φ A q c i = q d i q_{c_i}=q_{d_i} q c i = q d i i ∈ A i\in A i ∈ A A A A T A ≅ S L / J A T_A\cong S_L/J_A T A ≅ S L / J A m + 2 − ∣ A ∣ m+2-|A| m + 2 − ∣ A ∣ h h h L L L A A A J A : X L ∞ J_A : X_{\mathcal{L}}^\infty J A : X L ∞ Propositions 3.6 and 5.2.
Example 5.5. Consider four events 1 , 2 , 3 , 4 1,2,3,4 1 , 2 , 3 , 4 P P P { 1 , 2 , 3 , 4 } \{1,2,3,4\} { 1 , 2 , 3 , 4 } 1 < 2 1 < 2 1 < 2 1 < 4 1 < 4 1 < 4 3 < 4 3 < 4 3 < 4 L = J ( P ) L = \mathcal{J}(P) L = J ( P )
∅ ; { 1 } , { 3 } ; { 1 , 2 } , { 1 , 3 } ; { 1 , 2 , 3 } , { 1 , 3 , 4 } ; { 1 , 2 , 3 , 4 } , \varnothing;\quad\{1\},\ \{3\};\quad\{1,2\},\ \{1,3\};\quad\{1,2,3\},\ \{1,3,4\};\quad\{1,2,3,4\}, ∅ ; { 1 } , { 3 } ; { 1 , 2 } , { 1 , 3 } ; { 1 , 2 , 3 } , { 1 , 3 , 4 } ; { 1 , 2 , 3 , 4 } , listed by rank. Hence L L L T ( L ) = { 2 } \mathcal{T}(L) = \{2\} T ( L ) = { 2 } { 1 } ∨ { 3 } = { 1 , 3 } = { 1 , 2 , 3 } ∧ { 1 , 3 , 4 } \{1\} \vee\{3\} = \{1,3\} = \{1,2,3\} \wedge\{1,3,4\} { 1 } ∨ { 3 } = { 1 , 3 } = { 1 , 2 , 3 } ∧ { 1 , 3 , 4 } Figure 6 .
Figure 6. The thin distributive lattice L = J ( P ) L = \mathcal{J}(P) L = J ( P )
We write x ∅ , x 1 , x 3 , x 12 , … x_{\varnothing}, x_1, x_3, x_{12}, \ldots x ∅ , x 1 , x 3 , x 12 , … q U = ∏ p ∈ U θ p q_U = \prod_{p \in U} \theta_p q U = ∏ p ∈ U θ p q 134 = θ 1 θ 3 θ 4 q_{134} = \theta_1\theta_3\theta_4 q 134 = θ 1 θ 3 θ 4 θ 1 = θ 3 \theta_1 = \theta_3 θ 1 = θ 3 θ 2 = θ 3 \theta_2 = \theta_3 θ 2 = θ 3 θ 2 = θ 4 \theta_2 = \theta_4 θ 2 = θ 4 T ( L ) = { 2 } \mathcal{T}(L) = \{2\} T ( L ) = { 2 } [ 3 ] [3] [ 3 ] ∅ \varnothing ∅ { 1 } \{1\} { 1 } { 3 } \{3\} { 3 } A = { 1 } A = \{1\} A = { 1 } q 1 = q 3 q_1 = q_3 q 1 = q 3 q ∅ q 13 = q 1 2 q_{\varnothing}q_{13} = q_1^2 q ∅ q 13 = q 1 2 q 1 q 3 = q ∅ q 13 q_1q_3 = q_{\varnothing}q_{13} q 1 q 3 = q ∅ q 13
Now let A = { 2 } A = \{2\} A = { 2 } θ 2 = θ 3 \theta_2 = \theta_3 θ 2 = θ 3 q ∅ q 1234 = θ 1 θ 2 2 θ 4 = q 3 q 134 q_{\varnothing}q_{1234} = \theta_1\theta_2^2\theta_4 = q_3q_{134} q ∅ q 1234 = θ 1 θ 2 2 θ 4 = q 3 q 134 x ∅ x 1234 − x 3 x 134 x_{\varnothing}x_{1234} - x_3x_{134} x ∅ x 1234 − x 3 x 134 J A J_A J A Proposition 3.3 . Similarly, let A = { 1 , 3 } A = \{1,3\} A = { 1 , 3 } θ 1 = θ 3 \theta_1 = \theta_3 θ 1 = θ 3 θ 2 = θ 4 \theta_2 = \theta_4 θ 2 = θ 4 q ∅ q 1234 = θ 1 2 θ 2 2 = q 12 2 q_{\varnothing}q_{1234} = \theta_1^2\theta_2^2 = q_{12}^2 q ∅ q 1234 = θ 1 2 θ 2 2 = q 12 2 x ∅ x 1234 − x 12 2 ∉ J A x_{\varnothing}x_{1234} - x_{12}^2 \notin J_A x ∅ x 1234 − x 12 2 ∈ / J A Proposition 3.4 . In both cases, the join-meet relations and the linear relations q c i = q d i q_{c_i} = q_{d_i} q c i = q d i i ∈ A i \in A i ∈ A
(Takayuki Hibi) Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, The University of Osaka, Suita, Osaka 565-0871, Japan
Email address: hibi@math.sci.osaka-u.ac.jp
(Francesco Navarra) Sabanci University, Faculty of Engineering and Natural Sciences, Orta Mahalle, Tuzla 34956, Istanbul, Turkey
Email address: francesco.navarra@sabanciuniv.edu
(Ayesha Asloob Qureshi) Sabanci University, Faculty of Engineering and Natural Sciences, Orta Mahalle, Tuzla 34956, Istanbul, Turkey
Email address: aqureshi@sabanciuniv.edu, ayesha.asloob@sabanciuniv.edu
