Introduction
Let J κ ( x ) J_{\kappa}(x) J κ ( x ) κ \kappa κ κ ∈ C \kappa\in\mathbb{C} κ ∈ C κ ∈ R ≥ 0 \kappa\in\mathbb{R}_{\geq0} κ ∈ R ≥ 0 κ = m / 2 ∈ N ∪ ( N + 1 / 2 ) \kappa= m/2 \in\mathbb{N} \cup(\mathbb{N} + 1/2) κ = m /2 ∈ N ∪ ( N + 1/2 ) N = { 0 , 1 , 2 , … } \mathbb{N} = \{0, 1, 2, \ldots\} N = { 0 , 1 , 2 , … } R m \mathbb{R}^{m} R m
For κ ∈ R \kappa\in\mathbb{R} κ ∈ R J κ ( x ) J_{\kappa}(x) J κ ( x ) x = 0 x = 0 x = 0 [11 10.21(i)] ). We show that, for non-negative real κ ≠ 1 / 2 \kappa\ne1/2 κ = 1/2 n n n J κ ( x ) J_{\kappa}(x) J κ ( x ) V ⊂ ( C × ) n V \subset(\mathbb{C}^{\times})^{n} V ⊂ ( C × ) n V V V κ = 1 / 2 \kappa= 1/2 κ = 1/2 π Z \pi\mathbb{Z} π Z κ = − 1 / 2 \kappa= -1/2 κ = − 1/2 π ( Z + 1 / 2 ) \pi(\mathbb{Z} + 1/2) π ( Z + 1/2 )
Though J κ J_{\kappa} J κ x = 0 x = 0 x = 0 C × \mathbb{C}^{\times} C ×
1.1. Definition. Fix κ ∈ C \kappa\in\mathbb{C} κ ∈ C β ∈ C × \beta\in\mathbb{C}^{\times} β ∈ C × J κ J_{\kappa} J κ ( C × ) n (\mathbb{C}^{\times})^{n} ( C × ) n n n n ( β 1 , … , β n ) (\beta_{1}, \ldots, \beta_{n}) ( β 1 , … , β n ) ( C × ) n (\mathbb{C}^{\times})^{n} ( C × ) n x i = β i x_{i} = \beta_{i} x i = β i β i \beta_{i} β i x j = ± x k x_{j} = \pm x_{k} x j = ± x k j ≠ k j \ne k j = k
Fixing κ ∈ [ 0 , ∞ ) \kappa\in[0,\infty) κ ∈ [ 0 , ∞ ) [11 §10.21(vi)] ) which give that, as m → + ∞ m \to+\infty m → + ∞
β m = β κ , m = π ( m + κ / 2 − 1 / 4 ) + O κ ( m − 1 ) , \beta_m=\beta_{\kappa,m}=\pi\left(m+\kappa/2-1/4\right)+O_{\kappa}\left(m^{-1}\right), β m = β κ , m = π ( m + κ /2 − 1/4 ) + O κ ( m − 1 ) , where β κ , m \beta_{\kappa,m} β κ , m m m m J κ J_{\kappa} J κ m = 1 , 2 , … m=1,2,\ldots m = 1 , 2 , … height of β m \beta_m β m H ( β m ) = ∣ β m ∣ H(\beta_m)=|\beta_m| H ( β m ) = ∣ β m ∣ m m m κ \kappa κ T T T height , denoted H ( T ) H(T) H ( T ) k k k 2 k H k / π k 2^kH^k/\pi^k 2 k H k / π k H H H
By a maximal special subvariety T ⊂ V T \subset V T ⊂ V V V V V V V
Theorem 1.2. Let κ ∈ [ 0 , ∞ ) ∖ { 1 / 2 } \kappa\in[0,\infty)\setminus\{1/2\} κ ∈ [ 0 , ∞ ) ∖ { 1/2 } ϵ > 0 \epsilon>0 ϵ > 0 V ⊂ ( C × ) n V \subset(\mathbb{C}^{\times})^n V ⊂ ( C × ) n H ≥ 1 H \ge1 H ≥ 1 V V V
C ( V , κ , ϵ ) H ϵ C(V,\kappa,\epsilon)H^{\epsilon} C ( V , κ , ϵ ) H ϵ maximal Bessel special subvarieties T T T H ( T ) ≤ H H(T) \le H H ( T ) ≤ H κ ∈ N + 1 / 2 \kappa\in\mathbb{N}+1/2 κ ∈ N + 1/2 κ = ± 1 / 2 \kappa=\pm1/2 κ = ± 1/2
C ( V , κ ) ( log ( 1 + H ) ) c ( V , κ ) . C(V,\kappa)(\log(1+H))^{c(V,\kappa)}. C ( V , κ ) ( log ( 1 + H ) ) c ( V , κ ) . In particular, if V V V V V V H H H V ⊂ ( C × ) n V \subset(\mathbb{C}^{\times})^n V ⊂ ( C × ) n κ ∈ N + 1 / 2 \kappa\in\mathbb{N}+1/2 κ ∈ N + 1/2 κ ≠ 1 / 2 \kappa\ne1/2 κ = 1/2 [9 ] of Wilkie’s conjecture on point-counting bounds for sets definable using restricted Pfaffian functions and (unrestricted) exponential.
Theorem 1.2 is a weak analogue of the classical Manin-Mumford conjecture (theorem of Raynaud [29 ] ). It is weak in asserting only sparsity of the (maximal) special subvarieties in V V V [32 ] , the non-zero roots are transcendental when κ ∈ Q \kappa\in\mathbb{Q} κ ∈ Q
The proof of Theorem 1.2 uses point-counting and functional transcendence. It follows the basic strategy introduced in [27 ] , but with some key new features, including the manner in which special points are counted by integer points in a definable set. Definability of the counting set in an o-minimal structure entails that suitable related functions (essentially the Hankel functions with their oscillation “removed”) are so definable. Showing this uses summability theory of linear differential equations and quasi-analyticity; the relevant o-minimal structure is the structure R G , exp \mathbb{R}_{G,\exp} R G , e x p multi-summable Gevrey series (with exponentiation) introduced (and proved o-minimal) by van den Dries-Speissegger [12 ] .
Our functional transcendence result is an “Ax-Schanuel” statement for Bessel functions and exponentials. It is a generalization of the classical Ax-Schanuel theorem of Ax [3 Theorem 3] . A very special case is a result of Siegel [32 Satz 9] , proved in the context of the abovementioned transcendence results. This result was reproved by Kolchin [20 ] , who introduced some group-theoretic ideas that are central to our treatment (see [18 ] for a generalization). We combine Kolchin’s results and a study of the formal solutions and their Stokes phenomenon with the recent Ax-Schanuel technology [10 ] of Blázquez-Sanz–Casale–Freitag–Nagloo.
1.3. Definition. A (Bessel) weakly special subvariety of ( C × ) n (\mathbb{C}^{\times})^n ( C × ) n x i = c i x_i=c_i x i = c i c i ∈ C × c_i\in\mathbb{C}^{\times} c i ∈ C × x j = ± x k x_j=\pm x_k x j = ± x k j ≠ k j\ne k j = k
Let K K K D D D C \mathbb{C} C M M M L L L x 1 , … , x n x_1,\ldots,x_n x 1 , … , x n M M M y ‾ 1 , … , y ‾ m \overline{y}_1,\ldots,\overline{y}_m y 1 , … , y m M M M A 1 , … , A ℓ A_1,\ldots,A_\ell A 1 , … , A ℓ M M M A ‾ 1 , … , A ‾ k \overline{A}_1,\ldots,\overline{A}_k A 1 , … , A k
F = L ( x 1 , … , x n , y ‾ 1 , … , y ‾ m , A 1 , … , A ℓ , A ‾ 1 , … , A ‾ k , e t c ) F=L(x_1,\ldots,x_n,\overline{y}_1,\ldots,\overline{y}_m,A_1,\ldots,A_\ell,\overline{A}_1,\ldots,\overline{A}_k,\mathrm{etc}) F = L ( x 1 , … , x n , y 1 , … , y m , A 1 , … , A ℓ , A 1 , … , A k , etc ) the field obtained by adjoining to L L L td L F \operatorname{td}_L F td L F F / L F/L F / L
Bessel’s differential equation of order κ \kappa κ z = J κ ( x ) z=J_\kappa(x) z = J κ ( x )
x 2 z ′ ′ + x z ′ + ( x 2 − κ 2 ) z = 0 , x^2z''+xz'+(x^2-\kappa^2)z=0, x 2 z ′′ + x z ′ + ( x 2 − κ 2 ) z = 0 , where ′ ' ′ x x x generic if κ ∉ Z + 1 / 2 \kappa\notin\mathbb{Z}+1/2 κ ∈ / Z + 1/2 [20 ] ) to be S L 2 ( C ) \mathrm{SL}_2(\mathbb{C}) SL 2 ( C )
W = ( z 1 z 2 z 1 ′ z 2 ′ ) W=\begin{pmatrix} z_1 & z_2 \\ z'_1 & z'_2 \end{pmatrix} W = ( z 1 z 1 ′ z 2 z 2 ′ ) is a fundamental matrix formed from two C \mathbb{C} C z 1 , z 2 z_1,z_2 z 1 , z 2 td C ( x ) C ( x , W ) = 3 \operatorname{td}_{\mathbb{C}(x)}\mathbb{C}(x,W)=3 td C ( x ) C ( x , W ) = 3 κ \kappa κ e ± i x e^{\pm ix} e ± i x x x x td C ( x ) C ( x , W ) = 1 \operatorname{td}_{\mathbb{C}(x)}\mathbb{C}(x,W)=1 td C ( x ) C ( x , W ) = 1 K K K
Theorem 1.4. Let x 1 , … , x n ∈ K ∖ C x_1,\ldots,x_n\in K\setminus\mathbb{C} x 1 , … , x n ∈ K ∖ C i i i W i W_i W i x i x_i x i κ i ∈ C \kappa_i\in\mathbb{C} κ i ∈ C y 1 , … , y m ∈ K ∖ C y_1,\ldots,y_m\in K\setminus\mathbb{C} y 1 , … , y m ∈ K ∖ C j j j Y j Y_j Y j y j y_j y j ν j \nu_j ν j e 1 , … , e ℓ , t 1 , … , t ℓ ∈ K ∖ C e_1,\ldots,e_\ell,t_1,\ldots,t_\ell\in K\setminus\mathbb{C} e 1 , … , e ℓ , t 1 , … , t ℓ ∈ K ∖ C k k k D t k = D e k / e k Dt_k=De_k/e_k D t k = D e k / e k x ˉ = ( x 1 , … , x n ) \bar{x}=(x_1,\ldots,x_n) x ˉ = ( x 1 , … , x n ) W ‾ = ( W 1 , … , W n ) \overline{W}=(W_1,\ldots,W_n) W = ( W 1 , … , W n )
td C C ( t ˉ , e ˉ , y ˉ , Y ‾ , x ˉ , W ‾ ) ≥ ℓ + m + 3 n + 1 \operatorname{td}_{\mathbb{C}}\mathbb{C}( \bar{t},\bar{e},\bar{y},\overline{Y},\bar{x},\overline{W}) \ge\ell+m+3n+1 td C C ( t ˉ , e ˉ , y ˉ , Y , x ˉ , W ) ≥ ℓ + m + 3 n + 1 unless either
(A) t 1 , … , t ℓ , i y 1 , … , i y m t_1,\ldots,t_\ell,iy_1,\ldots,iy_m t 1 , … , t ℓ , i y 1 , … , i y m Q \mathbb{Q} Q C \mathbb{C} C
or
(B) there are i , j ≠ i i,j\ne i i , j = i x i = x j x_i=x_j x i = x j x i = − x j x_i=-x_j x i = − x j κ i ± κ j ∈ Z \kappa_i\pm\kappa_j\in\mathbb{Z} κ i ± κ j ∈ Z
We apply the above results to study sets Λ ⊂ R m \Lambda\subset\mathbb{R}^m Λ ⊂ R m Δ m ⊂ R m \Delta_m\subset\mathbb{R}^m Δ m ⊂ R m [14 , 15 , 17 , 36 ] . These are sets such that, for any λ 1 , λ 2 ∈ Λ \lambda_1,\lambda_2\in\Lambda λ 1 , λ 2 ∈ Λ λ 1 ≠ λ 2 \lambda_1\ne\lambda_2 λ 1 = λ 2
∫ Δ m exp ( 2 π i u ⋅ ( λ 1 − λ 2 ) ) d u = 0. \int_{\Delta_m}\exp\bigl(2\pi i u\cdot(\lambda_1-\lambda_2)\bigr)\,du=0. ∫ Δ m exp ( 2 π i u ⋅ ( λ 1 − λ 2 ) ) d u = 0. The orthogonality condition is equivalent to the distance between any two distinct points of Λ \Lambda Λ J m / 2 ( 2 π x ) J_{m/2}(2\pi x) J m /2 ( 2 π x ) Λ ∗ = 2 π Λ \Lambda^*=2\pi\Lambda Λ ∗ = 2 π Λ Bessel distance set in R m \mathbb{R}^m R m J m / 2 J_{m/2} J m /2
It is known that any such set is finite (Fuglede [14 ] states finiteness for m = 2 m=2 m = 2 [15 ] proves finiteness for all m ≥ 2 m\ge2 m ≥ 2 m m m m + 1 m+1 m + 1 [14 , 15 ] conjectured that such a set of size m + 2 m+2 m + 2 R m \mathbb{R}^m R m J m / 2 J_{m/2} J m /2
Iosevich and Kolountzakis [17 ] give a uniform linear bound for the size of such Λ \Lambda Λ m = 2 m=2 m = 2
inf { ∣ λ 1 − λ 2 ∣ : λ 1 , λ 2 ∈ Λ , λ 1 ≠ λ 2 } \inf\{|\lambda_1-\lambda_2|:\lambda_1,\lambda_2\in\Lambda,\lambda_1\ne\lambda_2\} inf { ∣ λ 1 − λ 2 ∣ : λ 1 , λ 2 ∈ Λ , λ 1 = λ 2 } in Λ \Lambda Λ
# Λ ∩ [ − R , R ] 2 ≤ c R 2 / 3 . \#\Lambda\cap[-R,R]^2\le cR^{2/3}. #Λ ∩ [ − R , R ] 2 ≤ c R 2/3 . The exponent 2 / 3 2/3 2/3 3 / 5 + ϵ 3/5+\epsilon 3/5 + ϵ [36 ] .
If we consider the tuple of distances associated with m + 2 m+2 m + 2 Λ \Lambda Λ V m + 2 , R m ⊂ R M V_{m+2,\mathbb{R}^{m}} \subset\mathbb{R}^{M} V m + 2 , R m ⊂ R M M = ( m + 2 ) ( m + 1 ) / 2 M=(m+2)(m+1)/2 M = ( m + 2 ) ( m + 1 ) /2 M M M m + 2 m+2 m + 2 R m \mathbb{R}^{m} R m R ( m + 2 ) ( m + 1 ) / 2 \mathbb{R}^{(m+2)(m+1)/2} R ( m + 2 ) ( m + 1 ) /2 [7 §9.7] ). Using 1.2 we prove
Theorem 1.5. Let m ≥ 2 m \ge2 m ≥ 2 ϵ > 0 \epsilon> 0 ϵ > 0 Λ ⊂ R m \Lambda\subset\mathbb{R}^{m} Λ ⊂ R m R m \mathbb{R}^{m} R m
# Λ ∩ [ − R , R ] m ≤ c ( m , ϵ ) R ϵ . \#\Lambda\cap[-R,R]^{m} \le c(m,\epsilon)R^{\epsilon}. #Λ ∩ [ − R , R ] m ≤ c ( m , ϵ ) R ϵ . For odd m m m C ( m ) ( log R ) c ( m ) C(m)(\log R)^{c(m)} C ( m ) ( log R ) c ( m )
Note that the constraints afforded by Theorem 1.4 are much less restrictive for non-generic orders than for generic ones. It is remarkable that, despite this, the same form holds for all non-negative real κ ≠ 1 / 2 \kappa\ne1/2 κ = 1/2 Theorem 1.2 (cf Lemma 5.1 versus Lemma 5.2 ), giving the same form of estimate in Theorem 1.5 .
The proof of Theorem 1.5 is in § 10 . Our present argument, relying on Theorem 1.2 , does not render effective constants.
In the final section § 11 of the paper we return to the question of possible algebraic relations among zeros of the Bessel functions relevant to Fuglede’s conjecture, and more generally those of rational order. For non-generic J m + 1 / 2 J_{m+1/2} J m + 1/2 [21 p30] , Conjecture 8.1) on the transcendence properties of the exponential function. We show that Schanuel’s conjecture implies the algebraic independence of distinct positive zeros and thus precludes the existence of a set of orthogonal exponentials in R m \mathbb{R}^{m} R m m + 2 m+2 m + 2 m ≥ 3 m \ge3 m ≥ 3
For even m = 2 k m=2k m = 2 k J k J_k J k m + 2 m+2 m + 2 R m \mathbb{R}^{m} R m [32 , 31 ] , we offer a formulation in Conjecture 8.4 .
Acknowledgments
Acknowledgments. GB was supported by the Marvin V. and Beverly J. Mielke Endowed Fund and the Infosys Member Fund as a member of the IAS. GB and AK were supported by the European Union (ERC, SharpOS, 101087910) and by Israel Science Foundation grant no. 2067/23.
JP is grateful to the Weizmann Institute of Science and the Institute for Advanced Study for their warm hospitality and financial support.
JP was supported as a Weston Visiting Professor during his visit to the the WIS, and by the Charles Simonyi Endowment as a member of the IAS.
GK was supported by the Israel Science Foundation and the Binational Science Foundation.
AI disclosure. GPT 5.6 sol and 6 Astra were used to check various details, including identifying some references, and to review an earlier fully human-written draft of this paper, leading to various suggested corrections that were human-implemented. In particular it spotted an error in an earlier version of the proof of Theorem 3.1 for rational non-integer parameters, and assisted in fixing this error. Note that this error did not affect our main applications for the Fuglede problem which deal with integer parameters.
Bessel Basics
Bessel’s differential equation (above) is of confluent hypergeometric type. It has an irregular singular point at x = ∞ x=\infty x = ∞ x = 0 x=0 x = 0
The Bessel function of the first kind , J κ ( x ) J_{\kappa}(x) J κ ( x )
J κ ( x ) = ( x 2 ) κ ∑ j = 0 ∞ ( − 1 ) j Γ ( j + 1 ) Γ ( j + κ + 1 ) ( x 2 ) 2 j . J_{\kappa}(x)=\left(\frac{x}{2}\right)^{\kappa}\sum_{j=0}^{\infty}\frac{(-1)^{j}}{\Gamma(j+1)\Gamma(j+\kappa+1)}\left(\frac{x}{2}\right)^{2j}. J κ ( x ) = ( 2 x ) κ j = 0 ∑ ∞ Γ ( j + 1 ) Γ ( j + κ + 1 ) ( − 1 ) j ( 2 x ) 2 j . (Note that Γ ( z ) − 1 \Gamma(z)^{-1} Γ ( z ) − 1
For integer κ \kappa κ J κ ( x ) J_{\kappa}(x) J κ ( x ) κ \kappa κ ( x / 2 ) κ (x/2)^{\kappa} ( x /2 ) κ J κ J_{\kappa} J κ
For non-integer κ \kappa κ C C C J − κ ( x ) J_{-\kappa}(x) J − κ ( x ) Bessel function of the second kind , Y κ Y_{\kappa} Y κ
Y κ ( x ) = J κ ( x ) cos ( π κ ) − J − κ ( x ) sin ( π κ ) . Y_{\kappa}(x)=\frac{J_{\kappa}(x)\cos(\pi\kappa)-J_{-\kappa}(x)}{\sin(\pi\kappa)}. Y κ ( x ) = sin ( π κ ) J κ ( x ) cos ( π κ ) − J − κ ( x ) . For integer κ \kappa κ J κ , J − κ J_{\kappa},J_{-\kappa} J κ , J − κ J κ = ( − 1 ) κ J − κ J_{\kappa}=(-1)^{\kappa}J_{-\kappa} J κ = ( − 1 ) κ J − κ Bessel function of the second kind Y κ Y_{\kappa} Y κ
Y κ ( x ) = lim α → κ Y α ( x ) , κ ∈ Z . Y_{\kappa}(x)=\lim_{\alpha\to\kappa}Y_{\alpha}(x),\qquad\kappa\in\mathbb{Z}. Y κ ( x ) = α → κ lim Y α ( x ) , κ ∈ Z . In all cases, J κ , Y κ J_{\kappa},Y_{\kappa} J κ , Y κ C C C κ \kappa κ
Wr ( J κ , Y κ ) = det ( J κ ( x ) Y κ ( x ) J κ ′ ( x ) Y κ ′ ( x ) ) = 2 π x . \operatorname{Wr}(J_{\kappa},Y_{\kappa})=\det\begin{pmatrix}J_{\kappa}(x)&Y_{\kappa}(x)\\J'_{\kappa}(x)&Y'_{\kappa}(x)\end{pmatrix}=\frac{2}{\pi x}. Wr ( J κ , Y κ ) = det ( J κ ( x ) J κ ′ ( x ) Y κ ( x ) Y κ ′ ( x ) ) = π x 2 . Finally, for all κ \kappa κ Hankel functions (also known as Bessel functions of the third kind ) are further canonical solutions of Bessel’s differential equation defined by
H κ ( 1 ) ( x ) = J κ ( x ) + i Y κ ( x ) , H κ ( 2 ) ( x ) = J κ ( x ) − i Y κ ( x ) . H_{\kappa}^{(1)}(x) = J_{\kappa}(x) + iY_{\kappa}(x), \qquad H_{\kappa}^{(2)}(x) = J_{\kappa}(x) - iY_{\kappa}(x). H κ ( 1 ) ( x ) = J κ ( x ) + i Y κ ( x ) , H κ ( 2 ) ( x ) = J κ ( x ) − i Y κ ( x ) . The principal branches have the asymptotic series
H κ ( 1 ) ( x ) = 2 π x e i ( x − κ π / 2 − π / 4 ) ∑ m = 0 ∞ ( 1 2 − κ ) m ( 1 2 + κ ) m m ! ( 2 i x ) m , H_{\kappa}^{(1)}(x) = \sqrt{\frac{2}{\pi x}} e^{i(x-\kappa\pi/2-\pi/4)} \sum_{m=0}^{\infty} \frac{\left(\frac{1}{2}-\kappa\right)_{m}\left(\frac{1}{2}+\kappa\right)_{m}}{m!(2ix)^{m}}, H κ ( 1 ) ( x ) = π x 2 e i ( x − κπ /2 − π /4 ) m = 0 ∑ ∞ m ! ( 2 i x ) m ( 2 1 − κ ) m ( 2 1 + κ ) m , H κ ( 2 ) ( x ) = 2 π x e − i ( x − κ π / 2 − π / 4 ) ∑ m = 0 ∞ ( 1 2 − κ ) m ( 1 2 + κ ) m m ! ( − 2 i x ) m , H_{\kappa}^{(2)}(x) = \sqrt{\frac{2}{\pi x}} e^{-i(x-\kappa\pi/2-\pi/4)} \sum_{m=0}^{\infty} \frac{\left(\frac{1}{2}-\kappa\right)_{m}\left(\frac{1}{2}+\kappa\right)_{m}}{m!(-2ix)^{m}}, H κ ( 2 ) ( x ) = π x 2 e − i ( x − κπ /2 − π /4 ) m = 0 ∑ ∞ m ! ( − 2 i x ) m ( 2 1 − κ ) m ( 2 1 + κ ) m , as ∣ x ∣ → ∞ |x| \to\infty ∣ x ∣ → ∞ arg x ∈ ( − π , 2 π ) \arg x \in(-\pi, 2\pi) arg x ∈ ( − π , 2 π ) arg x ∈ ( − 2 π , π ) \arg x \in(-2\pi, \pi) arg x ∈ ( − 2 π , π ) ( x ) m (x)_m ( x ) m
( x ) m = x ( x + 1 ) ⋯ ( x + m − 1 ) . (x)_m = x(x+1)\cdots(x+m-1). ( x ) m = x ( x + 1 ) ⋯ ( x + m − 1 ) . When κ ∉ Z + 1 / 2 \kappa\notin\mathbb{Z} + 1/2 κ ∈ / Z + 1/2 S L 2 ( C ) \mathrm{SL}_{2}(\mathbb{C}) SL 2 ( C ) [20 ] ).
See generally e.g. [5 , 11 , 25 ] .
Definability
For each fixed κ > − 1 2 \kappa> -\frac{1}{2} κ > − 2 1
f 1 ( x ) = x − 1 e − i x − 1 H κ ( 1 ) ( x − 1 ) , f 2 ( x ) = x − 1 e i x − 1 H κ ( 2 ) ( x − 1 ) , x > 0. f_{1}(x) = \sqrt{x^{-1}}e^{-ix^{-1}}H_{\kappa}^{(1)}(x^{-1}), \qquad f_{2}(x) = \sqrt{x^{-1}}e^{ix^{-1}}H_{\kappa}^{(2)}(x^{-1}), \qquad x > 0. f 1 ( x ) = x − 1 e − i x − 1 H κ ( 1 ) ( x − 1 ) , f 2 ( x ) = x − 1 e i x − 1 H κ ( 2 ) ( x − 1 ) , x > 0. 2.1. Lemma. The functions f 1 f_{1} f 1 f 2 f_{2} f 2 [ 0 , ∞ ) [0,\infty) [ 0 , ∞ ) 0 < ε < π / 2 0 < \varepsilon< \pi/2 0 < ε < π /2 C = C ( κ , ε ) > 0 C = C(\kappa,\varepsilon) > 0 C = C ( κ , ε ) > 0
∣ f j ( n ) ( z ) ∣ ≤ C n + 1 n ! 2 , ∀ n ∈ Z ≥ 0 , 0 < ∣ z ∣ ≤ 1 , ∣ arg z ∣ < π − ε , j = 1 , 2. \left|f_{j}^{(n)}(z)\right| \leq C^{n+1}n!^{2}, \qquad\forall n \in\mathbb{Z}_{\geq0},\ 0 < |z| \leq1,\ |\arg z| < \pi-\varepsilon,\ j = 1,2. f j ( n ) ( z ) ≤ C n + 1 n ! 2 , ∀ n ∈ Z ≥ 0 , 0 < ∣ z ∣ ≤ 1 , ∣ arg z ∣ < π − ε , j = 1 , 2. In particular, the real and imaginary parts of f 1 f_{1} f 1 f 2 f_{2} f 2 [ 0 , 1 ] [0,1] [ 0 , 1 ] R G \mathbb{R}_{G} R G [ 0 , ∞ ) [0,\infty) [ 0 , ∞ ) R G , exp \mathbb{R}_{G,\exp} R G , e x p [12 ] .
Holomorphic functions satisfying such derivative bounds on a bounded sector about the positive real axis of opening greater than π \pi π [12 ] ; see [6 , 13 ] (cf. [4 §4.5 E8] for another equivalent definition).
Proof. We begin with the integral identity, initially for positive real z z z
H κ ( 1 ) ( z ) = ( 2 π z ) 1 / 2 e i ( z − 1 2 π κ − 1 4 π ) Γ ( κ + 1 2 ) ∫ 0 ∞ e − u u κ − 1 2 ( 1 + i u 2 z ) κ − 1 2 d u H_{\kappa}^{(1)}(z)=\left(\frac{2}{\pi z}\right)^{1/2}\frac{e^{i(z-\frac{1}{2}\pi\kappa-\frac{1}{4}\pi)}}{\Gamma\left(\kappa+\frac{1}{2}\right)}\int_{0}^{\infty}e^{-u}u^{\kappa-\frac{1}{2}}\left(1+\frac{iu}{2z}\right)^{\kappa-\frac{1}{2}}\,du H κ ( 1 ) ( z ) = ( π z 2 ) 1/2 Γ ( κ + 2 1 ) e i ( z − 2 1 π κ − 4 1 π ) ∫ 0 ∞ e − u u κ − 2 1 ( 1 + 2 z i u ) κ − 2 1 d u which is stated in [2 §11.6] for the function K κ ( z ) = π 2 i κ + 1 H κ ( 1 ) ( i z ) K_{\kappa}(z)=\frac{\pi}{2}i^{\kappa+1}H_{\kappa}^{(1)}(iz) K κ ( z ) = 2 π i κ + 1 H κ ( 1 ) ( i z )
c κ = 2 π e − i ( π κ / 2 + π / 4 ) Γ ( κ + 1 2 ) , c_{\kappa}=\sqrt{\frac{2}{\pi}}\frac{e^{-i(\pi\kappa/2+\pi/4)}}{\Gamma\left(\kappa+\frac{1}{2}\right)}, c κ = π 2 Γ ( κ + 2 1 ) e − i ( π κ /2 + π /4 ) , we have
f 1 ( z ) = c κ ∫ 0 ∞ e − u u κ − 1 2 ( 1 + i u z 2 ) κ − 1 2 d u . f_{1}(z)=c_{\kappa}\int_{0}^{\infty}e^{-u}u^{\kappa-\frac{1}{2}}\left(1+\frac{iuz}{2}\right)^{\kappa-\frac{1}{2}}\,du. f 1 ( z ) = c κ ∫ 0 ∞ e − u u κ − 2 1 ( 1 + 2 i u z ) κ − 2 1 d u . If κ = m + 1 2 \kappa=m+\frac{1}{2} κ = m + 2 1 m ∈ Z ≥ 0 m\in\mathbb{Z}_{\geq0} m ∈ Z ≥ 0 z z z m m m κ ∉ Z + 1 2 \kappa\notin\mathbb{Z}+\frac{1}{2} κ ∈ / Z + 2 1
Since
d n d t n ( 1 + t ) κ − 1 2 = Γ ( κ + 1 2 ) Γ ( κ − n + 1 2 ) ( 1 + t ) κ − n − 1 2 , \frac{d^{n}}{dt^{n}}(1+t)^{\kappa-\frac{1}{2}}=\frac{\Gamma\left(\kappa+\frac{1}{2}\right)}{\Gamma\left(\kappa-n+\frac{1}{2}\right)}(1+t)^{\kappa-n-\frac{1}{2}}, d t n d n ( 1 + t ) κ − 2 1 = Γ ( κ − n + 2 1 ) Γ ( κ + 2 1 ) ( 1 + t ) κ − n − 2 1 , we get
f 1 ( n ) ( z ) = c κ Γ ( κ + 1 2 ) Γ ( κ − n + 1 2 ) i n 2 n ∫ 0 ∞ e − u u κ + n − 1 2 ( 1 + i u z 2 ) κ − n − 1 2 d u . f_{1}^{(n)}(z)=c_{\kappa}\frac{\Gamma\left(\kappa+\frac{1}{2}\right)}{\Gamma\left(\kappa-n+\frac{1}{2}\right)}\frac{i^{n}}{2^{n}}\int_{0}^{\infty}e^{-u}u^{\kappa+n-\frac{1}{2}}\left(1+\frac{iuz}{2}\right)^{\kappa-n-\frac{1}{2}}\,du. f 1 ( n ) ( z ) = c κ Γ ( κ − n + 2 1 ) Γ ( κ + 2 1 ) 2 n i n ∫ 0 ∞ e − u u κ + n − 2 1 ( 1 + 2 i u z ) κ − n − 2 1 d u . Fix 0 < ε < π / 2 0<\varepsilon<\pi/2 0 < ε < π /2
arg u = { − π 2 + ε 2 , 0 ≤ arg z < π − ε , π 2 − ε 2 , − π + ε < arg z < 0. \arg u=
\begin{cases}
-\frac{\pi}{2}+\frac{\varepsilon}{2}, & 0\leq\arg z<\pi-\varepsilon,\\
\frac{\pi}{2}-\frac{\varepsilon}{2}, & -\pi+\varepsilon<\arg z<0.
\end{cases} arg u = { − 2 π + 2 ε , 2 π − 2 ε , 0 ≤ arg z < π − ε , − π + ε < arg z < 0. These contour rotations are first performed for z z z u u u z z z
Re u = sin ( ε / 2 ) ∣ u ∣ , ε 2 ≤ arg ( i z u / 2 ) ≤ π − ε 2 . \operatorname{Re}u=\sin(\varepsilon/2)|u|,\qquad\frac{\varepsilon}{2}\leq\arg(izu/2)\leq\pi-\frac{\varepsilon}{2}. Re u = sin ( ε /2 ) ∣ u ∣ , 2 ε ≤ arg ( i z u /2 ) ≤ π − 2 ε . Thus the rotated integrals give analytic continuations into the indicated sectors, and
∣ 1 + i z u 2 ∣ ≥ sin ( ε / 2 ) . \left|1+\frac{izu}{2}\right|\geq\sin(\varepsilon/2). 1 + 2 i z u ≥ sin ( ε /2 ) . For integers n > κ − 1 2 n > \kappa- \frac{1}{2} n > κ − 2 1 C = C ( κ , ε ) > 0 C = C(\kappa,\varepsilon) > 0 C = C ( κ , ε ) > 0
∣ f 1 ( n ) ( z ) ∣ ≤ C n + 1 ∣ Γ ( κ + 1 2 ) Γ ( κ − n + 1 2 ) ∣ ∫ 0 ∞ e − t t κ + n − 1 2 d t = C n + 1 Γ ( κ + 1 2 ) ∣ Γ ( κ − n + 1 2 ) ∣ Γ ( κ + n + 1 2 ) . \begin{aligned}
\left|f_1^{(n)}(z)\right| \le C^{n+1} \left|\frac{\Gamma\left(\kappa+\frac{1}{2}\right)}{\Gamma\left(\kappa-n+\frac{1}{2}\right)}\right| \int_0^\infty e^{-t}t^{\kappa+n-\frac{1}{2}}\,dt \\
&= C^{n+1}\frac{\Gamma\left(\kappa+\frac{1}{2}\right)}{\left|\Gamma\left(\kappa-n+\frac{1}{2}\right)\right|}\Gamma\left(\kappa+n+\frac{1}{2}\right).
\end{aligned} f 1 ( n ) ( z ) ≤ C n + 1 Γ ( κ − n + 2 1 ) Γ ( κ + 2 1 ) ∫ 0 ∞ e − t t κ + n − 2 1 d t = C n + 1 Γ ( κ − n + 2 1 ) Γ ( κ + 2 1 ) Γ ( κ + n + 2 1 ) . By the reflection formula for the Gamma function and Stirling’s approximation, for these n n n C > 0 C > 0 C > 0
Γ ( κ + n + 1 2 ) ∣ Γ ( κ − n + 1 2 ) ∣ = ∣ cos ( π κ ) ∣ π Γ ( n + κ + 1 2 ) Γ ( n − κ + 1 2 ) ≤ C n + 1 n ! 2 . \frac{\Gamma\left(\kappa+n+\frac{1}{2}\right)}{\left|\Gamma\left(\kappa-n+\frac{1}{2}\right)\right|} = \frac{|\cos(\pi\kappa)|}{\pi}\Gamma\left(n+\kappa+\frac{1}{2}\right)\Gamma\left(n-\kappa+\frac{1}{2}\right) \le C^{n+1}n!^2. Γ ( κ − n + 2 1 ) Γ ( κ + n + 2 1 ) = π ∣ cos ( π κ ) ∣ Γ ( n + κ + 2 1 ) Γ ( n − κ + 2 1 ) ≤ C n + 1 n ! 2 . For the finitely many integers 0 ≤ n ≤ κ − 1 2 0 \le n \le\kappa-\frac{1}{2} 0 ≤ n ≤ κ − 2 1 ∣ z ∣ ≤ 1 |z| \le1 ∣ z ∣ ≤ 1 ∣ 1 + i z u / 2 ∣ ≤ 1 + ∣ u ∣ / 2 |1+izu/2| \le1+|u|/2 ∣1 + i z u /2∣ ≤ 1 + ∣ u ∣/2 C C C n n n f 2 f_2 f 2 H κ ( 2 ) ( z ) = H κ ( 1 ) ( z ‾ ) H_\kappa^{(2)}(z)=H_\kappa^{(1)}(\overline{z}) H κ ( 2 ) ( z ) = H κ ( 1 ) ( z ) x → 0 + x \to0^+ x → 0 +
The sectorial estimates imply that the real and imaginary parts of f 1 f_1 f 1 f 2 f_2 f 2 R G \mathbb R_{\mathcal G} R G 0 0 0 0 0 0 [ 0 , 1 ] [0,1] [ 0 , 1 ] x ≥ 1 x \ge1 x ≥ 1 t = 1 / x ∈ ( 0 , 1 ] t=1/x\in(0,1] t = 1/ x ∈ ( 0 , 1 ] J κ ( t ) J_\kappa(t) J κ ( t ) Y κ ( t ) Y_\kappa(t) Y κ ( t ) [ 0 , 1 ] [0,1] [ 0 , 1 ] t t t κ \kappa κ log t \log t log t t α = exp ( α log t ) t^\alpha=\exp(\alpha\log t) t α = exp ( α log t ) R a n , exp \mathbb R_{\mathrm{an},\exp} R an , e x p t 1 / 2 e ∓ i t t^{1/2}e^{\mp it} t 1/2 e ∓ i t f 1 f_1 f 1 f 2 f_2 f 2 [ 1 , ∞ ) [1,\infty) [ 1 , ∞ ) R a n , exp ⊆ R G , exp \mathbb R_{\mathrm{an},\exp}\subseteq\mathbb R_{\mathcal G,\exp} R an , e x p ⊆ R G , e x p □ \square □
Recurrence relation
Bessel functions satisfy the following recurrence relations (see e.g. [11 §10.61] )
J κ + 1 ( x ) = 2 κ x J κ ( x ) − J κ − 1 ( x ) , J κ + 1 ′ ( x ) = − 2 J κ ′ ( x ) + J κ − 1 ′ ( x ) . J_{\kappa+1}(x)=\frac{2\kappa}{x}J_\kappa(x)-J_{\kappa-1}(x), \qquad J_{\kappa+1}'(x)=-2J_\kappa'(x)+J_{\kappa-1}'(x). J κ + 1 ( x ) = x 2 κ J κ ( x ) − J κ − 1 ( x ) , J κ + 1 ′ ( x ) = − 2 J κ ′ ( x ) + J κ − 1 ′ ( x ) . Eliminating J κ − 1 J_{\kappa-1} J κ − 1 κ , κ + 1 \kappa,\kappa+1 κ , κ + 1
J κ + 1 ( x ) = κ x J κ ( x ) − J κ ′ ( x ) . J_{\kappa+1}(x)=\frac{\kappa}{x}J_\kappa(x)-J_\kappa'(x). J κ + 1 ( x ) = x κ J κ ( x ) − J κ ′ ( x ) . Differentiating and using the differential equation will give a similar identity for J κ + 1 ′ ( x ) J_{\kappa+1}'(x) J κ + 1 ′ ( x ) J κ ( x ) J_\kappa(x) J κ ( x ) J κ ′ ( x ) J_\kappa'(x) J κ ′ ( x ) [11 §10.61] the same identities hold for Y κ ( x ) Y_{\kappa}(x) Y κ ( x ) H κ ( i ) ( x ) H_{\kappa}^{(i)}(x) H κ ( i ) ( x ) C \mathbb{C} C
Non-generic order
When κ = m + 1 / 2 ∈ Z + 1 / 2 \kappa=m+1/2\in\mathbb{Z}+1/2 κ = m + 1/2 ∈ Z + 1/2 e ± i x e^{\pm ix} e ± i x x \sqrt{x} x m m m [11 10.49(i) via 10.47(ii)] )
H m + 1 / 2 ( 1 ) = ( − i ) m + 1 2 π x e i x ∑ k = 0 m c ( m , k ) ( i x ) k H_{m+1/2}^{(1)}=(-i)^{m+1}\sqrt{\frac{2}{\pi x}}e^{ix}\sum_{k=0}^{m}c(m,k)\left(\frac{i}{x}\right)^{k} H m + 1/2 ( 1 ) = ( − i ) m + 1 π x 2 e i x k = 0 ∑ m c ( m , k ) ( x i ) k H m + 1 / 2 ( 2 ) = i m + 1 2 π x e − i x ∑ k = 0 m c ( m , k ) ( − i x ) k H_{m+1/2}^{(2)}=i^{m+1}\sqrt{\frac{2}{\pi x}}e^{-ix}\sum_{k=0}^{m}c(m,k)\left(\frac{-i}{x}\right)^{k} H m + 1/2 ( 2 ) = i m + 1 π x 2 e − i x k = 0 ∑ m c ( m , k ) ( x − i ) k where c ( m , k ) = ( m + k ) ! / ( ( m − k ) ! k ! 2 k ) c(m,k)=(m+k)!/((m-k)!k!2^{k}) c ( m , k ) = ( m + k )! / (( m − k )! k ! 2 k )
H m + 1 / 2 ( 1 ) = J m + 1 / 2 + i Y m + 1 / 2 , H m + 1 / 2 ( 2 ) = J m + 1 / 2 − i Y m + 1 / 2 . H_{m+1/2}^{(1)}=J_{m+1/2}+iY_{m+1/2},\qquad H_{m+1/2}^{(2)}=J_{m+1/2}-iY_{m+1/2}. H m + 1/2 ( 1 ) = J m + 1/2 + i Y m + 1/2 , H m + 1/2 ( 2 ) = J m + 1/2 − i Y m + 1/2 . For negative non-generic orders one has
H − m − 1 / 2 ( 1 ) = i ( − 1 ) m H m + 1 / 2 ( 1 ) , H − m − 1 / 2 ( 2 ) = − i ( − 1 ) m H m + 1 / 2 ( 2 ) . H_{-m-1/2}^{(1)}=i(-1)^{m}H_{m+1/2}^{(1)},\qquad H_{-m-1/2}^{(2)}=-i(-1)^{m}H_{m+1/2}^{(2)}. H − m − 1/2 ( 1 ) = i ( − 1 ) m H m + 1/2 ( 1 ) , H − m − 1/2 ( 2 ) = − i ( − 1 ) m H m + 1/2 ( 2 ) . Ax-Lindemann for generic Bessel equations
This section proves a result on the algebraic independence of logarithms and solutions of Bessel’s differential equation with several different generic orders on algebraically dependent arguments. The initial result, for two algebraically dependent arguments, is the key for the general Bessel-Ax-Schanuel statement.
Let x , y x,y x , y X X X C ( X ) = C ( x , y ) \mathbb{C}(X)=\mathbb{C}(x,y) C ( X ) = C ( x , y ) K = C ( x , y ) K=\mathbb{C}(x,y) K = C ( x , y ) D = d / d x D=d/dx D = d / d x
We consider the Bessel d.e. on the x x x κ \kappa κ y y y λ \lambda λ y y y
Let M / K M/K M / K C \mathbb{C} C L x , L y L_x,L_y L x , L y L x / K L_x/K L x / K K K K z 1 ( x ) , z 2 ( x ) z_1(x),z_2(x) z 1 ( x ) , z 2 ( x ) κ \kappa κ L y / K L_y/K L y / K K K K w 1 ( y ) , w 2 ( y ) w_1(y), w_2(y) w 1 ( y ) , w 2 ( y ) λ \lambda λ
td K L x = td K L y = 3. \operatorname{td}_{K}L_x=\operatorname{td}_{K}L_y=3. td K L x = td K L y = 3. Let L = L x L y L=L_xL_y L = L x L y L x , L y , L L_x,L_y,L L x , L y , L K K K
Theorem 3.1. In the above setting, suppose that
td K L < 6. \operatorname{td}_{K}L<6. td K L < 6. Then either y = x y=x y = x κ ± λ ∈ Z \kappa\pm\lambda\in\mathbb{Z} κ ± λ ∈ Z y = − x y=-x y = − x κ ± λ ∈ Z \kappa\pm\lambda\in\mathbb{Z} κ ± λ ∈ Z
Proof. By the results of Kolchin [20 italicized conclusion at the bottom of p 1158] , if td K L < 6 \operatorname{td}_{K}L<6 td K L < 6 A ∈ GL 2 ( K ) A\in\operatorname{GL}_2(K) A ∈ GL 2 ( K ) α ∈ L \alpha\in L α ∈ L α 2 ∈ K \alpha^2\in K α 2 ∈ K
W y = α A W x C W_y=\alpha A W_x C W y = α A W x C holds between any pair of fundamental matrices W y W_y W y y y y W x W_x W x x x x C ∈ GL 2 ( C ) C\in\operatorname{GL}_2(\mathbb{C}) C ∈ GL 2 ( C )
We take the canonical Hankel function solutions on P 1 ( C ) \mathbb{P}^1(\mathbb{C}) P 1 ( C ) W x , W y W_x,W_y W x , W y x , y x,y x , y X X X X X X
Suppose P ∈ X P\in X P ∈ X Poleset ( y ) \operatorname{Poleset}(y) Poleset ( y ) P ∈ Poleset ( x ) P\in\operatorname{Poleset}(x) P ∈ Poleset ( x ) y y y Poleset ( y ) = Poleset ( x ) = Poleset \operatorname{Poleset}(y)=\operatorname{Poleset}(x)=\operatorname{Poleset} Poleset ( y ) = Poleset ( x ) = Poleset
Let P ∈ Poleset P\in\operatorname{Poleset} P ∈ Poleset y y y x x x y y y x x x
y = c m x m + c m − 1 / k x m − 1 / k + ⋯ + c 1 x 1 / k + regular terms y=c_mx^m+c_{m-1/k}x^{m-1/k}+\cdots+c_1x^{1/k}+\text{regular terms} y = c m x m + c m − 1/ k x m − 1/ k + ⋯ + c 1 x 1/ k + regular terms where m ∈ Q m\in\mathbb{Q} m ∈ Q k k k ∞ \infty ∞ x − 1 / k x^{-1/k} x − 1/ k P P P Reg P ( x ) \operatorname{Reg}_P(x) Reg P ( x )
We next consider the exponential factors in the formal asymptotic expansions of the Hankel function solutions at ∞ \infty ∞ P P P
y = ± x + Reg P ( x ) . y=\pm x+\operatorname{Reg}_P(x). y = ± x + Reg P ( x ) . Thus, in particular, x x x y y y P ∈ Poleset P\in\operatorname{Poleset} P ∈ Poleset y = ± x + Reg Q ( x ) y=\pm x+\operatorname{Reg}_{Q}(x) y = ± x + Reg Q ( x ) Q ∈ Poleset Q\in\operatorname{Poleset} Q ∈ Poleset y ∓ x y\mp x y ∓ x X = P 1 X=\mathbb{P}^{1} X = P 1 x x x y y y y = ± x y=\pm x y = ± x
In this case, the conclusion that λ ± κ ∈ Z \lambda\pm\kappa\in\mathbb{Z} λ ± κ ∈ Z [32 ] (and was reproved by Kolchin [20 ] ). We give another argument.
First, we show that in the relation W y = α A W x C W_y=\alpha A W_x C W y = α A W x C α ∈ C ( x ) \alpha\in\mathbb{C}(x) α ∈ C ( x ) α A = W y C − 1 W x − 1 \alpha A=W_y C^{-1}W_x^{-1} α A = W y C − 1 W x − 1 α \alpha α { 0 , ∞ } \{0,\infty\} { 0 , ∞ } ∞ \infty ∞ x − 1 / 2 x^{-1/2} x − 1/2 α \alpha α α 2 \alpha^{2} α 2 x = 0 x=0 x = 0 α \alpha α C ( x ) \mathbb{C}(x) C ( x )
Now suppose κ ∉ Z \kappa\notin\mathbb{Z} κ ∈ / Z W x W_x W x J κ ( x ) , J − κ ( x ) J_{\kappa}(x),J_{-\kappa}(x) J κ ( x ) , J − κ ( x ) x = 0 x=0 x = 0 e ( κ ) , e ( − κ ) e(\kappa),e(-\kappa) e ( κ ) , e ( − κ ) W y W_y W y 0 0 0 { e ( ± λ ) } = { e ( ± κ ) } \{e(\pm\lambda)\}=\{e(\pm\kappa)\} { e ( ± λ )} = { e ( ± κ )} λ ± κ ∈ Z \lambda\pm\kappa\in\mathbb{Z} λ ± κ ∈ Z κ , λ \kappa,\lambda κ , λ λ ∓ κ ∈ Z \lambda\mp\kappa\in\mathbb{Z} λ ∓ κ ∈ Z
Thus to prove the theorem it remains to show that the sign in y = ± x + Reg Q ( x ) y=\pm x+\operatorname{Reg}_{Q}(x) y = ± x + Reg Q ( x ) Q ∈ Poleset Q\in\operatorname{Poleset} Q ∈ Poleset
So let us suppose, as we may, that there is P ∈ Poleset P\in\operatorname{Poleset} P ∈ Poleset y = + x + Reg P ( x ) y=+x+\operatorname{Reg}_{P}(x) y = + x + Reg P ( x ) W y , W x W_y,W_x W y , W x W ~ y , W ~ x \widetilde{W}_y,\widetilde{W}_x W y , W x
W ~ y = α A W ~ x C \widetilde{W}_y=\alpha A\widetilde{W}_xC W y = α A W x C is preserved by the summation process. The exponential torus
T μ = ( μ 0 0 μ − 1 ) , μ ∈ C × T_{\mu}=
\begin{pmatrix}
\mu& 0\\
0 & \mu^{-1}
\end{pmatrix},
\qquad\mu\in\mathbb{C}^{\times} T μ = ( μ 0 0 μ − 1 ) , μ ∈ C × acts by
W ~ y ↦ W ~ y T μ , W ~ x ↦ W ~ x T μ , \widetilde{W}_y\mapsto\widetilde{W}_yT_{\mu},\qquad\widetilde{W}_x\mapsto\widetilde{W}_xT_{\mu}, W y ↦ W y T μ , W x ↦ W x T μ , and this implies that C C C T μ T_{\mu} T μ
Now suppose Q ∈ Poleset Q\in\operatorname{Poleset} Q ∈ Poleset Q = P Q=P Q = P
W y = α A W x C W_y=\alpha A W_xC W y = α A W x C to Q Q Q x x x Q Q Q W x W_x W x y y y x → ∞ x\to\infty x → ∞ m m m π i \pi i π i m ∈ Z m\in\mathbb{Z} m ∈ Z y = ± x + Reg Q ( x ) y = \pm x + \operatorname{Reg}_Q(x) y = ± x + Reg Q ( x ) Q Q Q m m m
We work on the Riemann surface of the logarithm and write y = y ∗ e m π i y = y^*e^{m\pi i} y = y ∗ e mπ i y ∗ y^* y ∗ y ∗ = x + Reg Q ( x ) y^* = x + \operatorname{Reg}_Q(x) y ∗ = x + Reg Q ( x )
We now apply the “analytic continuation” (with reflection) formula from [11 §10.11] :
W z e m π i = N m W z S ν , m W_{ze^{m\pi i}} = N_m W_z S_{\nu,m} W z e mπ i = N m W z S ν , m for the Hankel functions of order ν \nu ν
N m = ( 1 0 0 e m π i ) , N_m = \begin{pmatrix} 1 & 0 \\ 0 & e^{m\pi i} \end{pmatrix}, N m = ( 1 0 0 e mπ i ) , S ν , m = 1 sin ( ν π ) ( − sin ( ( m − 1 ) ν π ) e ν π i sin ( m ν π ) − e − ν π i sin ( m ν π ) sin ( ( m + 1 ) ν π ) ) , S_{\nu,m} = \frac{1}{\sin(\nu\pi)}
\begin{pmatrix}
-\sin((m-1)\nu\pi) & e^{\nu\pi i}\sin(m\nu\pi) \\
-e^{-\nu\pi i}\sin(m\nu\pi) & \sin((m+1)\nu\pi)
\end{pmatrix}, S ν , m = sin ( ν π ) 1 ( − sin (( m − 1 ) ν π ) − e − ν π i sin ( m ν π ) e ν π i sin ( m ν π ) sin (( m + 1 ) ν π ) ) , and where the limiting values are taken for integral ν \nu ν ν ∈ Z \nu\in\mathbb{Z} ν ∈ Z
S ν , m = ( − ( − 1 ) m ν ( m − 1 ) ( − 1 ) ( m − 1 ) ν e ν π i m − ( − 1 ) ( m − 1 ) ν e − ν π i m ( − 1 ) m ν ( m + 1 ) ) . S_{\nu,m} =
\begin{pmatrix}
-(-1)^{m\nu}(m-1) & (-1)^{(m-1)\nu}e^{\nu\pi i m} \\
-(-1)^{(m-1)\nu}e^{-\nu\pi i m} & (-1)^{m\nu}(m+1)
\end{pmatrix}. S ν , m = ( − ( − 1 ) m ν ( m − 1 ) − ( − 1 ) ( m − 1 ) ν e − ν π im ( − 1 ) ( m − 1 ) ν e ν π im ( − 1 ) m ν ( m + 1 ) ) . We have
W y ∗ e m π i = N m W y ∗ S λ , m = α A W x C , W_{y^*e^{m\pi i}} = N_m W_{y^*}S_{\lambda,m} = \alpha A W_x C, W y ∗ e mπ i = N m W y ∗ S λ , m = α A W x C , so that
N m W ~ y ∗ S λ , m = α A W ~ x C , N_m \widetilde{W}_{y^*}S_{\lambda,m} = \alpha A\widetilde{W}_x C, N m W y ∗ S λ , m = α A W x C , and, under the exponential torus,
N m W ~ y ∗ T μ S λ , m = α A W ~ x T μ C . N_m \widetilde{W}_{y^*}T_\mu S_{\lambda,m} = \alpha A\widetilde{W}_xT_\mu C. N m W y ∗ T μ S λ , m = α A W x T μ C . Comparing the last two relations, and since C − 1 T μ C = T μ C^{-1}T_\mu C = T_\mu C − 1 T μ C = T μ
S λ , m − 1 T μ S λ , m = C − 1 T μ C = T μ . S_{\lambda,m}^{-1}T_\mu S_{\lambda,m} = C^{-1}T_\mu C = T_\mu. S λ , m − 1 T μ S λ , m = C − 1 T μ C = T μ . This entails that S λ , m S_{\lambda,m} S λ , m
If λ \lambda λ m = 0 m = 0 m = 0 y = + x + Reg Q ( x ) y = +x + \operatorname{Reg}_Q(x) y = + x + Reg Q ( x ) λ ∈ Z \lambda\in\mathbb{Z} λ ∈ Z m = 0 m = 0 m = 0
So we are left with λ = a / b ∈ Q \lambda= a/b \in\mathbb{Q} λ = a / b ∈ Q a , b a,b a , b b > 2 b > 2 b > 2 λ ∈ Z + 1 / 2 \lambda\in\mathbb{Z} + 1/2 λ ∈ Z + 1/2 S λ , m S_{\lambda,m} S λ , m sin ( m λ π ) = 0 \sin(m\lambda\pi) = 0 sin ( mλπ ) = 0 b ∣ m b\mid m b ∣ m
Let us consider the (possibly multivalued) function
ϕ = ( y x ) 2 / b \phi= \left(\frac{y}{x}\right)^{2/b} ϕ = ( x y ) 2/ b on X X X ϕ ( P ) = 1 \phi(P) = 1 ϕ ( P ) = 1 ϕ \phi ϕ y / x y/x y / x
As any path from P P P Q Q Q Q Q Q b ∣ m b\mid m b ∣ m ϕ \phi ϕ Q Q Q Q Q Q Q Q Q ϕ \phi ϕ Q Q Q Q ∈ Poleset Q \in\operatorname{Poleset} Q ∈ Poleset ϕ \phi ϕ X X X
Since b ∣ m b \mid m b ∣ m ϕ \phi ϕ 1 1 1 Q ∈ Poleset Q \in\operatorname{Poleset} Q ∈ Poleset Q Q Q x , y x,y x , y k k k ϕ \phi ϕ 1 1 1 Q Q Q k k k ϕ \phi ϕ 1 1 1 N N N N N N x x x y y y x , y x,y x , y ϕ \phi ϕ y y y ϕ \phi ϕ 2 N / b 2N/b 2 N / b
Hence ϕ \phi ϕ y = x y=x y = x
The transcendence indeed drops in the exceptional cases.
Next we consider non-constant rational functions
e 1 , … , e m , x 1 , … , x n e_1,\ldots,e_m,x_1,\ldots,x_n e 1 , … , e m , x 1 , … , x n on a smooth projective complex curve X X X C ( X ) = C ( e i , x j ) \mathbb{C}(X)=\mathbb{C}(e_i,x_j) C ( X ) = C ( e i , x j ) t j = log e j t_j=\log e_j t j = log e j W i W_i W i κ i \kappa_i κ i x i x_i x i
3.2. Theorem. In the above setting, suppose that
td C ( x 1 ) ( e ‾ , t ‾ , x ‾ , W ‾ ) < m + 3 n . \operatorname{td}_{\mathbb{C}(x_1)}(\overline{e},\overline{t},\overline{x},\overline{W}) < m+3n. td C ( x 1 ) ( e , t , x , W ) < m + 3 n . Then either (A) or (B) holds where:
(A) The t j t_j t j Q \mathbb{Q} Q C \mathbb{C} C
(B) There exists distinct indices j , k j,k j , k
(i) x k = x j x_k=x_j x k = x j κ j ± κ k ∈ Z \kappa_j \pm\kappa_k \in\mathbb{Z} κ j ± κ k ∈ Z
(ii) x k = − x j x_k=-x_j x k = − x j κ j ± κ k ∈ Z \kappa_j \pm\kappa_k \in\mathbb{Z} κ j ± κ k ∈ Z
Proof. By Kolchin’s results [20: Theorem p 1152, Ostrowski’s theorem p 1155, and italicized text bottom p 1158] applied to the Picard-Vessiot extension of C ( X ) \mathbb{C}(X) C ( X )
(a) a C \mathbb{C} C C \mathbb{C} C t j t_j t j [3 ] this must be a Q \mathbb{Q} Q C \mathbb{C} C
or
(b) a pair i , j i,j i , j i ≠ j i \ne j i = j Theorem 3.1 . This leads to (B). □
Weakly special subvarieties and Ax-Schanuel
We start with the main result of [10 ] . The setting is a principal G G G P → Y P \to Y P → Y Y Y Y G G G
Theorem. ([10 Theorem A] ) Let ∇ \nabla ∇ G G G P → Y P \to Y P → Y Gal ( ∇ ) = G \operatorname{Gal}(\nabla)=G Gal ( ∇ ) = G V V V P P P L \mathcal{L} L A ⊂ V ∩ L A \subset V \cap\mathcal{L} A ⊂ V ∩ L P P P
If dim V < dim G + dim A \dim V < \dim G+\dim A dim V < dim G + dim A A A A Y Y Y ∇ \nabla ∇
We refer to [10 Lemma 2.3] for the definition of the Galois group Gal ( ∇ ) \operatorname{Gal}(\nabla) Gal ( ∇ ) ∇ \nabla ∇ X ⊂ Y X \subset Y X ⊂ Y Gal ( ∇ ∣ X ) \operatorname{Gal}(\nabla|_{X}) Gal ( ∇ ∣ X ) Gal ( ∇ ) \operatorname{Gal}(\nabla) Gal ( ∇ ) [10 Definition 3.4 (and informally at p1)] for the definition of a “sparse” algebraic group. We note that powers of G m \mathbb{G}_{m} G m G a \mathbb{G}_{a} G a S L 2 \mathrm{SL}_{2} SL 2
We consider solutions of a system of logarithm and Bessel differential equations. We take m m m d t j d e j = 1 e j \frac{dt_j}{de_j}=\frac{1}{e_j} d e j d t j = e j 1 n n n κ 1 , … , κ n \kappa_1,\ldots,\kappa_n κ 1 , … , κ n x 1 , … , x n x_1,\ldots,x_n x 1 , … , x n G = G a ( C ) m × S L 2 ( C ) n G=\mathbb{G}_{a}(\mathbb{C})^{m}\times\mathrm{SL}_{2}(\mathbb{C})^{n} G = G a ( C ) m × SL 2 ( C ) n G G G P → Y P \to Y P → Y G G G ∇ \nabla ∇ Y = ( C × ) m × ( C × ) n Y=(\mathbb{C}^{\times})^{m}\times(\mathbb{C}^{\times})^{n} Y = ( C × ) m × ( C × ) n t ˉ = ( t 1 , … , t m ) \bar{t}=(t_1,\ldots,t_m) t ˉ = ( t 1 , … , t m ) h ˉ = ( h 1 , … , h n ) \bar{h}=(h_1,\ldots,h_n) h ˉ = ( h 1 , … , h n )
P = { ( t ˉ , h ˉ , e ˉ , x ˉ ) ∈ G a ( C ) m × G L 2 ( C ) n × ( C × ) m × ( C × ) n : P=\left\{(\bar{t},\bar{h},\bar{e},\bar{x})\in\mathbb{G}_{a}(\mathbb{C})^{m}\times\mathrm{GL}_{2}(\mathbb{C})^{n}\times(\mathbb{C}^{\times})^{m}\times(\mathbb{C}^{\times})^{n}:\right. P = { ( t ˉ , h ˉ , e ˉ , x ˉ ) ∈ G a ( C ) m × GL 2 ( C ) n × ( C × ) m × ( C × ) n : det ( h j ) = 2 π x j , j = 1 , … , n } . \left.\det(h_j)=\frac{2}{\pi x_j},\ j=1,\ldots,n\right\}. det ( h j ) = π x j 2 , j = 1 , … , n } . Locally on Y Y Y m + 3 n m+3n m + 3 n C ( Y ) \mathbb{C}(Y) C ( Y )
To get an explicit Ax-Schanuel statement we must explicate the maximal weakly special subvarieties of Y Y Y W ⊂ Y W \subset Y W ⊂ Y G G G [10 Examples 2.1 and 2.5] ).
Proposition. In this setting, the maximal weakly special subvarieties W ⊂ Y W \subset Y W ⊂ Y
(i) W W W e 1 k 1 ⋯ e m k m = c e_1^{k_1}\cdots e_m^{k_m}=c e 1 k 1 ⋯ e m k m = c k i ∈ Z k_i \in\mathbb{Z} k i ∈ Z c ∈ C × c \in\mathbb{C}^{\times} c ∈ C ×
(ii) W W W x i = c x_i=c x i = c i i i c ∈ C × c \in\mathbb{C}^{\times} c ∈ C ×
(iii) W W W x j = x k x_j=x_k x j = x k k , j ≠ k k,j\ne k k , j = k κ k ± κ j ∈ Z \kappa_k\pm\kappa_j\in\mathbb{Z} κ k ± κ j ∈ Z
(iv) W W W x j = − x k x_j=-x_k x j = − x k k , j ≠ k k,j\ne k k , j = k κ k ± κ j ∈ Z \kappa_k\pm\kappa_j\in\mathbb{Z} κ k ± κ j ∈ Z
Proof. If a subvariety W ⊂ Y W\subset Y W ⊂ Y C C C
To see this, suppose W ⊂ Y W\subset Y W ⊂ Y k _k k e 1 k 1 ⋯ e m k m = c e_1^{k_1}\cdots e_m^{k_m}=c e 1 k 1 ⋯ e m k m = c k i ∈ Z k_i\in\mathbb{Z} k i ∈ Z c ∈ C × c\in\mathbb{C}^{\times} c ∈ C × i _i i x i = c x_i=c x i = c c ∈ C × c\in\mathbb{C}^{\times} c ∈ C × j k _{jk} j k x j = x k x_j=x_k x j = x k j ≠ k j\ne k j = k κ k ± κ j ∈ Z \kappa_k\pm\kappa_j\in\mathbb{Z} κ k ± κ j ∈ Z j k _{jk} j k x j = − x k x_j=-x_k x j = − x k j ≠ k j\ne k j = k κ k ± κ j ∈ Z \kappa_k\pm\kappa_j\in\mathbb{Z} κ k ± κ j ∈ Z
The claim is that W W W
To prove the claim, let P ∈ W P\in W P ∈ W W W W P P P
So we can find a direction avoiding all these conditions, and find an algebraic curve C ⊂ W C\subset W C ⊂ W P P P
Now Theorem 4.1 gives a complex form of Ax-Schanuel for Bessel and logarithm functions. It is then a standard matter to deduce a differential field version of Ax-Schanuel from the complex version. We can include Bessel d.e. of non-generic order since C ( e i y , i y ) \mathbb{C}(e^{iy},iy) C ( e i y , i y ) C ( y , W ( y ) ) \mathbb{C}(y,W(y)) C ( y , W ( y )) W ( y ) W(y) W ( y )
Proof of Theorem 1.4
First, we get rid of the non-generic Bessel d.e. by including further logarithm d.e, and renumber so that we work with e j e_j e j t j t_j t j x j x_j x j κ j \kappa_j κ j e j e_j e j x j x_j x j A A A P P P V V V A A A A A A V V V
td C ( t ˉ , e ˉ , x ˉ , W ˉ ) < ℓ + 3 n + 1 \operatorname{td}_{\mathbb{C}}\left(\bar{t},\bar{e},\bar{x},\bar{W}\right)<\ell+3n+1 td C ( t ˉ , e ˉ , x ˉ , W ˉ ) < ℓ + 3 n + 1 then (as rank A = 1 \operatorname{rank} A=1 rank A = 1 dim G = ℓ + 3 n \dim G=\ell+3n dim G = ℓ + 3 n
dim V < dim G + dim A \dim V<\dim G+\dim A dim V < dim G + dim A and we conclude that the locus of e i , x j e_i,x_j e i , x j C \mathbb{C} C
One could also give a version with several commuting derivations.
Two Lemmas
In this section, T = { ( t 1 ( s ) , … , t n ( s ) ) : s ∈ Δ } T=\{(t_1(s),\ldots,t_n(s)):s\in\Delta\} T = {( t 1 ( s ) , … , t n ( s )) : s ∈ Δ } t i ( s ) t_i(s) t i ( s ) Δ ⊂ C \Delta\subset\mathbb{C} Δ ⊂ C s = 0 s=0 s = 0 T T T t i ( s ) t_i(s) t i ( s ) s ∈ Δ s\in\Delta s ∈ Δ
The proofs involve only local arguments (in particular, appeals to Ax-Schanuel), except that in the proof of Lemma 5.2 we eventually reduce to global considerations concerning a certain rational function.
Generic Lemma
Let κ ∈ C \kappa\in\mathbb{C} κ ∈ C V ⊂ ( C × ) n V\subset(\mathbb{C}^{\times})^n V ⊂ ( C × ) n T = ( t 1 ( s ) , … , t n ( s ) ) T=(t_1(s),\ldots,t_n(s)) T = ( t 1 ( s ) , … , t n ( s )) s ∈ Δ s\in\Delta s ∈ Δ C n \mathbb{C}^n C n x ˉ ( s ) ∈ V \bar{x}(s)\in V x ˉ ( s ) ∈ V x ˉ ( s ) = ( x 1 ( s ) , … , x n ( s ) ) \bar{x}(s)=(x_1(s),\ldots,x_n(s)) x ˉ ( s ) = ( x 1 ( s ) , … , x n ( s )) s ∈ Δ s\in\Delta s ∈ Δ
e − 2 π i t j H κ ( 1 ) ( x j ) + e 2 π i t j H κ ( 2 ) ( x j ) = 0 , j = 1 , … , n . e^{-2\pi i t_j}H_{\kappa}^{(1)}(x_j)+e^{2\pi i t_j}H_{\kappa}^{(2)}(x_j)=0,\qquad j=1,\ldots,n. e − 2 π i t j H κ ( 1 ) ( x j ) + e 2 π i t j H κ ( 2 ) ( x j ) = 0 , j = 1 , … , n . Then there is a weakly special subvariety W ⊂ V W\subset V W ⊂ V { x ˉ ( s ) } \{\bar{x}(s)\} { x ˉ ( s )}
Proof. We suppose V ⊂ ( C × ) n V\subset(\mathbb{C}^{\times})^n V ⊂ ( C × ) n W = V = ( C × ) n W=V=(\mathbb{C}^{\times})^n W = V = ( C × ) n
2 π i t j , e 2 π i t j , x j , W j , 2\pi i t_j,\qquad e^{2\pi i t_j},\qquad x_j,\qquad W_j, 2 π i t j , e 2 π i t j , x j , W j , where W j W_j W j κ \kappa κ
1 + n + dim V + 2 n ≤ n + 3 n 1+n+\dim V+2n\le n+3n 1 + n + dim V + 2 n ≤ n + 3 n (as dim V < n \dim V<n dim V < n x 1 , … , x k x_1,\ldots,x_k x 1 , … , x k n ′ = n − k n'=n-k n ′ = n − k x j x_j x j t j t_j t j m ′ ′ m'' m ′′ t j t_j t j Q \mathbb{Q} Q C \mathbb{C} C n ′ ′ n'' n ′′ x x x x j ≠ ± x i x_j\ne\pm x_i x j = ± x i W W W x i x_i x i V ′ ′ ⊂ W ∩ V V'' \subset W \cap V V ′′ ⊂ W ∩ V x ˉ ( s ) \bar{x}(s) x ˉ ( s ) dim W = n ′ ′ \dim W = n'' dim W = n ′′ ± \pm ± x j x_j x j t j t_j t j
1 + m ′ ′ + dim V ′ ′ + 2 n ′ ′ 1 + m'' + \dim V'' + 2n'' 1 + m ′′ + dim V ′′ + 2 n ′′ If dim V ′ ′ < dim W = n ′ ′ \dim V'' < \dim W = n'' dim V ′′ < dim W = n ′′ W W W W ⊂ V W \subset V W ⊂ V { x ˉ ( s ) } ⊂ W \{\bar{x}(s)\} \subset W { x ˉ ( s )} ⊂ W
Now we establish the corresponding result for a non-generic order κ \kappa κ
5.2. Non-Generic Lemma. Let κ ≠ ± 1 / 2 \kappa\ne\pm1/2 κ = ± 1/2 V ⊂ ( C × ) n V \subset(\mathbb{C}^{\times})^n V ⊂ ( C × ) n T = ( t 1 ( s ) , … , t n ( s ) ) \mathbf{T} = (t_1(s), \ldots, t_n(s)) T = ( t 1 ( s ) , … , t n ( s )) s ∈ Δ s \in\Delta s ∈ Δ C n \mathbb{C}^n C n x ˉ ( s ) ∈ V \bar{x}(s) \in V x ˉ ( s ) ∈ V x ˉ ( s ) = ( x 1 ( s ) , … , x n ( s ) ) \bar{x}(s) = (x_1(s), \ldots, x_n(s)) x ˉ ( s ) = ( x 1 ( s ) , … , x n ( s )) s ∈ Δ s \in\Delta s ∈ Δ
e − 2 π i t j H κ ( 1 ) ( x j ) + e 2 π i t j H κ ( 2 ) ( x j ) = 0 , j = 1 , … , n . e^{-2\pi i t_j} H_{\kappa}^{(1)}(x_j) + e^{2\pi i t_j} H_{\kappa}^{(2)}(x_j) = 0,\qquad j = 1,\ldots,n. e − 2 π i t j H κ ( 1 ) ( x j ) + e 2 π i t j H κ ( 2 ) ( x j ) = 0 , j = 1 , … , n . Then there is a weakly special subvariety W ⊂ V W \subset V W ⊂ V { x ˉ ( s ) } \{\bar{x}(s)\} { x ˉ ( s )}
Proof. As in 5.1 we may suppose V V V κ \kappa κ V V V
e 4 π i t j = R ( x j ) e 2 i x j e^{4\pi i t_j} = R(x_j)e^{2ix_j} e 4 π i t j = R ( x j ) e 2 i x j for j = 1 , … , n j = 1,\ldots,n j = 1 , … , n R R R R = R κ R = R_\kappa R = R κ κ = m + 1 / 2 \kappa= m + 1/2 κ = m + 1/2 m m m
H κ ( 1 ) ( x ) H κ ( 2 ) ( x ) = − R ( x ) e 2 i x , \frac{H_{\kappa}^{(1)}(x)}{H_{\kappa}^{(2)}(x)} = -R(x)e^{2ix}, H κ ( 2 ) ( x ) H κ ( 1 ) ( x ) = − R ( x ) e 2 i x , specifically (see §2)
R = ( − 1 ) m ∑ k = 0 m c ( m , k ) ( i / x ) k ∑ k = 0 m c ( m , k ) ( − i / x ) k . R = (-1)^m \frac{\sum_{k=0}^{m} c(m,k)(i/x)^k}{\sum_{k=0}^{m} c(m,k)(-i/x)^k}. R = ( − 1 ) m ∑ k = 0 m c ( m , k ) ( − i / x ) k ∑ k = 0 m c ( m , k ) ( i / x ) k . We observe that R R R κ ≠ ± 1 / 2 \kappa\ne\pm1/2 κ = ± 1/2 R × \mathbb{R}^{\times} R × R ( x ) R ( − x ) = 1 R(x)R(-x) = 1 R ( x ) R ( − x ) = 1
Now some x j x_j x j t j t_j t j V V V V ⊂ ( C × ) n V \subset(\mathbb{C}^{\times})^n V ⊂ ( C × ) n
Let us suppose that the Q \mathbb{Q} Q C \mathbb{C} C { 4 π i t j } \{4\pi i t_j\} { 4 π i t j } d d d Q \mathbb{Q} Q C \mathbb{C} C { 4 π i t j , 2 i x h } \{4\pi i t_j,2ix_h\} { 4 π i t j , 2 i x h } d + δ d+\delta d + δ x j x_j x j ( C × ) n (\mathbb{C}^{\times})^n ( C × ) n k k k k ≤ n − 1 k\le n-1 k ≤ n − 1 V V V
Write y j = R ( x j ) y_j=R(x_j) y j = R ( x j )
log y j = 4 π i t j − 2 i x j . \log y_j=4\pi i t_j-2ix_j. log y j = 4 π i t j − 2 i x j . Note also that, since R R R td C ( y ‾ ) = td C ( x ‾ ) = k \operatorname{td}_{\mathbb{C}}(\overline{y})=\operatorname{td}_{\mathbb{C}}(\overline{x})=k td C ( y ) = td C ( x ) = k r r r { y j } \{y_j\} { y j }
By Ax-Schanuel, taking logarithms locally, we have
1 + r ≤ td C ( log y ‾ , y ‾ ) . 1+r\le\operatorname{td}_{\mathbb{C}}\left(\log\overline{y},\overline{y}\right). 1 + r ≤ td C ( log y , y ) . The t j t_j t j 1 + k 1+k 1 + k t j t_j t j C ( x ‾ ) \mathbb{C}(\overline{x}) C ( x ) k k k r < k r<k r < k t j t_j t j C ( x ‾ ) \mathbb{C}(\overline{x}) C ( x ) r = k r=k r = k
Now we observe that
td C ( 4 π i t ‾ , exp ( 4 π i t ‾ ) , 2 i x ‾ , exp ( 2 i x ‾ ) ) ≤ min ( 1 + δ + d + 0 , 1 + k + d + 0 ) \operatorname{td}_{\mathbb{C}}\left(4\pi i\overline{t},\exp(4\pi i\overline{t}),2i\overline{x},\exp(2i\overline{x})\right)\le\min(1+\delta+d+0,1+k+d+0) td C ( 4 π i t , exp ( 4 π i t ) , 2 i x , exp ( 2 i x ) ) ≤ min ( 1 + δ + d + 0 , 1 + k + d + 0 ) (as each exp ( 2 i x j ) \exp(2ix_j) exp ( 2 i x j ) x j , exp ( 4 π i t j ) x_j,\exp(4\pi i t_j) x j , exp ( 4 π i t j )
1 + δ ≤ td C ( 4 π i t ‾ , 2 i x ‾ ) ≤ 1 + δ + d − d 1+\delta\le\operatorname{td}_{\mathbb{C}}\left(4\pi i\overline{t},2i\overline{x}\right)\le1+\delta+d-d 1 + δ ≤ td C ( 4 π i t , 2 i x ) ≤ 1 + δ + d − d and we have equality. But this transcendence degree is 1 + k 1+k 1 + k k = δ k=\delta k = δ
Now suppose (after renumbering) that 2 i x 1 , … , 2 i x k 2ix_1,\ldots,2ix_k 2 i x 1 , … , 2 i x k Q \mathbb{Q} Q 4 π i t j 4\pi i t_j 4 π i t j 2 i x h 2ix_h 2 i x h Q \mathbb{Q} Q 2 i x 1 , … , 2 i x k 2ix_1,\ldots,2ix_k 2 i x 1 , … , 2 i x k 4 π i t j 4\pi i t_j 4 π i t j t j t_j t j x ‾ \overline{x} x Q \mathbb{Q} Q 2 i x 1 , … , 2 i x k 2ix_1,\ldots,2ix_k 2 i x 1 , … , 2 i x k
Therefore, all the algebraic relations (and there is at least one) defining the Zariski closure of the locus { x ‾ } \{\overline{x}\} { x } ( C × ) n (\mathbb{C}^{\times})^n ( C × ) n Q \mathbb{Q} Q C \mathbb{C} C x 1 , … , x k x_1,\ldots,x_k x 1 , … , x k
x h = ∑ q h j x j + c h , h = k + 1 , … , n x_h=\sum q_{hj}x_j+c_h,\qquad h=k+1,\ldots,n x h = ∑ q hj x j + c h , h = k + 1 , … , n where q h j ∈ Q q_{hj}\in\mathbb{Q} q hj ∈ Q c h ∈ C c_h\in\mathbb{C} c h ∈ C
Now also y 1 , … , y k y_1,\ldots,y_k y 1 , … , y k C \mathbb{C} C C \mathbb{C} C
y h α h h = b h ∏ j = 1 k y j α h j , h = k + 1 , … , n , y_h^{\alpha_{hh}} = b_h \prod_{j=1}^{k} y_j^{\alpha_{hj}}, \qquad h = k+1,\ldots,n, y h α hh = b h j = 1 ∏ k y j α hj , h = k + 1 , … , n , where α h h ≠ 0 \alpha_{hh} \ne0 α hh = 0 α h j ∈ Z \alpha_{hj} \in\mathbb{Z} α hj ∈ Z j = 1 , … , k j = 1,\ldots,k j = 1 , … , k b h ∈ C b_h \in\mathbb{C} b h ∈ C h = k + 1 , … , n h = k+1,\ldots,n h = k + 1 , … , n
R ( ∑ j = 1 k q h j x j + c h ) α h h = b h ∏ j = 1 k R ( x j ) α h j . R\left(\sum_{j=1}^{k} q_{hj}x_j+c_h\right)^{\alpha_{hh}} = b_h \prod_{j=1}^{k} R(x_j)^{\alpha_{hj}}. R ( j = 1 ∑ k q hj x j + c h ) α hh = b h j = 1 ∏ k R ( x j ) α hj . As R R R x 1 , … , x k x_1,\ldots,x_k x 1 , … , x k h h h α h ℓ \alpha_{h\ell} α h ℓ α h ℓ = ± α h h \alpha_{h\ell} = \pm\alpha_{hh} α h ℓ = ± α hh
∑ j = 1 k q h j x j + c h = q h ℓ x ℓ + c . \sum_{j=1}^{k} q_{hj}x_j+c_h = q_{h\ell}x_\ell+c. j = 1 ∑ k q hj x j + c h = q h ℓ x ℓ + c . So we are reduced to considering (dropping subscripts, and according as α = ± 1 \alpha= \pm1 α = ± 1
R ( q x + c ) = b R ( x ) , R ( q x + c ) R ( x ) = b R(qx+c)=bR(x), \qquad R(qx+c)R(x)=b R ( q x + c ) = b R ( x ) , R ( q x + c ) R ( x ) = b We consider the first (α = 1 \alpha=1 α = 1
Suppose q = 1 q=1 q = 1 c ≠ 0 c\ne0 c = 0 c = 0 c=0 c = 0 b = 1 b=1 b = 1
Suppose q ≠ 1 q\ne1 q = 1 x ↦ q x + c x\mapsto qx+c x ↦ q x + c ξ = x − δ \xi=x-\delta ξ = x − δ ξ \xi ξ ξ ↦ q ξ + ( q δ + c − δ ) \xi\mapsto q\xi+(q\delta+c-\delta) ξ ↦ q ξ + ( q δ + c − δ ) q ≠ 1 q\ne1 q = 1 δ \delta δ ξ \xi ξ ξ ↦ q ξ \xi\mapsto q\xi ξ ↦ q ξ R ∗ ( q ξ ) = b R ∗ ( ξ ) R^*(q\xi)=bR^*(\xi) R ∗ ( q ξ ) = b R ∗ ( ξ ) R ∗ ( ξ ) = R ( ξ + δ ) R^*(\xi)=R(\xi+\delta) R ∗ ( ξ ) = R ( ξ + δ ) R ∗ R^* R ∗
This leaves us to consider q = − 1 q=-1 q = − 1 x ↦ − x + c x\mapsto-x+c x ↦ − x + c b 2 = 1 b^2=1 b 2 = 1 b = ± 1 b=\pm1 b = ± 1 R ( x ) R ( − x ) = 1 R(x)R(-x)=1 R ( x ) R ( − x ) = 1
R ( − x + c ) = b R ( x ) = b / R ( − x ) therefore R ( − x ) R ( − x + c ) = b R(-x+c)=bR(x)=b/R(-x) \qquad\text{therefore} \qquad R(-x)R(-x+c)=b R ( − x + c ) = b R ( x ) = b / R ( − x ) therefore R ( − x ) R ( − x + c ) = b which we may rewrite as
R ( x ) R ( x + c ) = b = ± 1. R(x)R(x+c)=b=\pm1. R ( x ) R ( x + c ) = b = ± 1. Now if x 0 x_0 x 0 R R R x 0 + c x_0+c x 0 + c x 0 + c x_0+c x 0 + c x 0 + 2 c x_0+2c x 0 + 2 c c ≠ 0 c\ne0 c = 0 c = 0 c=0 c = 0 R R R
So for α = 1 \alpha= 1 α = 1 q = 1 q = 1 q = 1 c = 0 c = 0 c = 0 b = 1 b = 1 b = 1 x h = x j x_h = x_j x h = x j j = 1 , … , k j = 1,\ldots,k j = 1 , … , k
Now we consider α = − 1 \alpha= -1 α = − 1 R ( q x + c ) R ( x ) = b R(qx+c)R(x) = b R ( q x + c ) R ( x ) = b q = 1 q = 1 q = 1 q ≠ ± 1 q \ne\pm1 q = ± 1 x ↦ q x + c x \mapsto qx+c x ↦ q x + c q ∈ Q q \in\mathbb{Q} q ∈ Q
So we consider q = − 1 q = -1 q = − 1 R ( x ) R ( − x ) = 1 R(x)R(-x) = 1 R ( x ) R ( − x ) = 1 c ≠ 0 c \ne0 c = 0
It isn’t, as the orbits under the two maps are infinite.
So for α = − 1 \alpha= -1 α = − 1 R ( x ) R ( − x ) = 1 R(x)R(-x) = 1 R ( x ) R ( − x ) = 1 x h = − x j x_h = -x_j x h = − x j j = 1 , … , k j = 1,\ldots,k j = 1 , … , k
And such a relation must occur unless V V V x i x_i x i
Repeating on the new weakly special we eventually find a positive-dimensional weakly special subvariety W ⊂ V W \subset V W ⊂ V { x ‾ ( s ) } \{\overline{x}(s)\} { x ( s )}
Bessel-Manin-Mumford
Proof of Theorem 1.2
Let κ ∈ [ 0 , ∞ ) ∖ { 1 / 2 } \kappa\in[0,\infty)\setminus\{1/2\} κ ∈ [ 0 , ∞ ) ∖ { 1/2 } V ⊂ ( C × ) n V \subset(\mathbb{C}^{\times})^n V ⊂ ( C × ) n
We can deal with the finitely many non-real zeros by a simple reduction after which it suffices to consider non-zero real zeros only (which, as has been noted, are symmetrically located about x = 0 x = 0 x = 0
We prove the theorem by induction on the ambient dimension n n n n = 1 n = 1 n = 1 V ⊂ ( C × ) n V \subset(\mathbb{C}^{\times})^n V ⊂ ( C × ) n U ⊂ ( C × ) m U \subset(\mathbb{C}^{\times})^m U ⊂ ( C × ) m m < n m < n m < n
The weakly special subvarieties in ( C × ) n (\mathbb{C}^{\times})^n ( C × ) n V V V T T T m ( T ) < n m(T) < n m ( T ) < n V V V V T ⊂ ( C × ) m ( T ) V_T \subset(\mathbb{C}^{\times})^{m(T)} V T ⊂ ( C × ) m ( T ) T T T V V V V T V_T V T V T V_T V T
The union of all positive-dimensional weakly special subvarieties of V V V V p w s p V^{\mathrm{pwsp}} V pwsp V − V p w s p V - V^{\mathrm{pwsp}} V − V pwsp
Z = Z n , κ ( V ) = { ( t 1 , … , t n ) ∈ R n : ∃ x 1 , … , x n ∈ ( R × ) n : Z=Z_{n,\kappa}(V)=\{(t_1,\ldots,t_n)\in\mathbb{R}^n:\exists x_1,\ldots,x_n\in(\mathbb{R}^{\times})^n: Z = Z n , κ ( V ) = {( t 1 , … , t n ) ∈ R n : ∃ x 1 , … , x n ∈ ( R × ) n : ( x 1 , … , x n ) ∈ V − V p w s p , 0 ≤ ∣ x j ∣ − 2 π t j < 2 π , j = 1 , … , n , and (x_1,\ldots,x_n)\in V-V^{\mathrm{pwsp}},\qquad0\leq|x_j|-2\pi t_j<2\pi,\quad j=1,\ldots,n,\quad\text{and} ( x 1 , … , x n ) ∈ V − V pwsp , 0 ≤ ∣ x j ∣ − 2 π t j < 2 π , j = 1 , … , n , and e − 2 π i t j H κ ( 1 ) ( ∣ x j ∣ ) + e 2 π i t j H κ ( 2 ) ( ∣ x j ∣ ) = 0 , j = 1 , … , n } . e^{-2\pi i t_j}H_{\kappa}^{(1)}(|x_j|)+e^{2\pi i t_j}H_{\kappa}^{(2)}(|x_j|)=0,\quad j=1,\ldots,n\}. e − 2 π i t j H κ ( 1 ) ( ∣ x j ∣ ) + e 2 π i t j H κ ( 2 ) ( ∣ x j ∣ ) = 0 , j = 1 , … , n } . Given t ˉ ∈ Z \bar{t}\in Z t ˉ ∈ Z x ˉ \bar{x} x ˉ
Any locally defined positive-dimensional semi-algebraic curve in Z Z Z ( t 1 ( s ) , … , t n ( s ) ) (t_1(s),\ldots,t_n(s)) ( t 1 ( s ) , … , t n ( s )) s ∈ I s\in I s ∈ I x j ( s ) , s ∈ I x_j(s),s\in I x j ( s ) , s ∈ I R a n \mathbb{R}_{\mathrm{an}} R an x ( t ) x(t) x ( t ) Δ \Delta Δ t i ( s ) t_i(s) t i ( s ) x i ( s ) x_i(s) x i ( s ) ∣ x j ∣ = σ j x j |x_j|=\sigma_jx_j ∣ x j ∣ = σ j x j σ j ∈ { ± 1 } \sigma_j\in\{\pm1\} σ j ∈ { ± 1 } ( ∣ x 1 ∣ , … , ∣ x n ∣ ) (|x_1|,\ldots,|x_n|) ( ∣ x 1 ∣ , … , ∣ x n ∣ ) W W W V σ ˉ = { ( x 1 , … , x n ) : ( σ 1 x 1 , … , σ n x n ) ∈ V } V^{\bar{\sigma}}=\{(x_1,\ldots,x_n):(\sigma_1x_1,\ldots,\sigma_nx_n)\in V\} V σ ˉ = {( x 1 , … , x n ) : ( σ 1 x 1 , … , σ n x n ) ∈ V } x ˉ \bar{x} x ˉ W σ ˉ ⊂ V W^{\bar{\sigma}}\subset V W σ ˉ ⊂ V
Now we apply block counting to Z Z Z V V V c ( Z , ϵ ) T ϵ c(Z,\epsilon)T^\epsilon c ( Z , ϵ ) T ϵ V V V W ⊂ V W\subset V W ⊂ V
There are at most c ( Z , ϵ ) T ϵ c(Z,\epsilon)T^\epsilon c ( Z , ϵ ) T ϵ V V V
When κ \kappa κ κ ≠ 1 / 2 \kappa\neq1/2 κ = 1/2 [9 ] . □
Remark
6.2. Remark. We note that all constants are uniform for V V V [8 ] where a fewnomial differential elimination result is required as well.
Bessel distance sets and Fuglede problem
Up to dilation, the Fuglede problem concerns sets Λ ⊂ R m \Lambda\subset\mathbb{R}^m Λ ⊂ R m J m / 2 J_{m/2} J m /2 Λ ⊂ R m \Lambda\subset\mathbb{R}^{m} Λ ⊂ R m J κ J_{\kappa} J κ κ \kappa κ m / 2 m/2 m /2 κ = ± 1 / 2 \kappa=\pm1/2 κ = ± 1/2 Theorem 1.5 .
Theorem 7.1. Let κ ∈ R + \kappa\in\mathbb{R}_{+} κ ∈ R + κ ≠ 1 / 2 \kappa\ne1/2 κ = 1/2 m ≥ 1 m \ge1 m ≥ 1 ϵ > 0 \epsilon> 0 ϵ > 0 Λ ⊂ R m \Lambda\subset\mathbb{R}^{m} Λ ⊂ R m J κ J_{\kappa} J κ C = C ( κ , m , ϵ ) C=C(\kappa,m,\epsilon) C = C ( κ , m , ϵ ) R ≥ 1 R \ge1 R ≥ 1
# Λ ∩ [ − R , R ] m ≤ C R ϵ \#\Lambda\cap[-R,R]^{m} \le CR^{\epsilon} #Λ ∩ [ − R , R ] m ≤ C R ϵ If κ = r + 1 / 2 \kappa=r+1/2 κ = r + 1/2 r ≥ 1 r \ge1 r ≥ 1
# Λ ∩ [ − R , R ] m ≤ C ( log R ) c \#\Lambda\cap[-R,R]^{m} \le C(\log R)^{c} #Λ ∩ [ − R , R ] m ≤ C ( log R ) c for suitable constants C = C ( m , κ ) C=C(m,\kappa) C = C ( m , κ ) c = c ( m , κ ) c=c(m,\kappa) c = c ( m , κ ) R ≥ e R \ge e R ≥ e
Given positive integers k k k m m m distance variety V k , R m V_{k,\mathbb{R}^{m}} V k , R m R k ( k − 1 ) / 2 \mathbb{R}^{k(k-1)/2} R k ( k − 1 ) /2
( R m ) k ⟶ R k ( k − 1 ) / 2 , (\mathbb{R}^{m})^{k} \longrightarrow\mathbb{R}^{k(k-1)/2}, ( R m ) k ⟶ R k ( k − 1 ) /2 , ( P 1 , … , P k ) ⟼ ( d ( P i , P j ) : 1 ≤ i < j ≤ k ) (P_{1},\ldots,P_{k}) \longmapsto(d(P_{i},P_{j}):1\le i<j\le k) ( P 1 , … , P k ) ⟼ ( d ( P i , P j ) : 1 ≤ i < j ≤ k ) on points P 1 , … , P k ∈ R m P_{1},\ldots,P_{k}\in\mathbb{R}^{m} P 1 , … , P k ∈ R m ( i , j ) (i,j) ( i , j ) V V V k ≥ m + 2 k\ge m+2 k ≥ m + 2 k = m + 2 k=m+2 k = m + 2 Cayley-Menger determinant
CM m ( d ‾ ) = ( 0 d 12 2 d 13 2 ⋯ d 1 k 2 1 d 12 2 0 d 23 2 ⋯ d 2 k 2 1 d 13 2 d 23 2 0 ⋱ d 3 k 2 1 ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ d 1 k 2 d 2 k 2 d 3 k 2 ⋯ 0 1 1 1 1 1 1 0 ) \operatorname{CM}_{m}(\overline{d})=
\begin{pmatrix}
0 & d_{12}^{2} & d_{13}^{2} & \cdots& d_{1k}^{2} & 1 \\
d_{12}^{2} & 0 & d_{23}^{2} & \cdots& d_{2k}^{2} & 1 \\
d_{13}^{2} & d_{23}^{2} & 0 & \ddots& d_{3k}^{2} & 1 \\
\vdots& \vdots& \vdots& \ddots& \vdots& \vdots\\
d_{1k}^{2} & d_{2k}^{2} & d_{3k}^{2} & \cdots& 0 & 1 \\
1 & 1 & 1 & 1 & 1 & 0
\end{pmatrix} CM m ( d ) = 0 d 12 2 d 13 2 ⋮ d 1 k 2 1 d 12 2 0 d 23 2 ⋮ d 2 k 2 1 d 13 2 d 23 2 0 ⋮ d 3 k 2 1 ⋯ ⋯ ⋱ ⋱ ⋯ 1 d 1 k 2 d 2 k 2 d 3 k 2 ⋮ 0 1 1 1 1 ⋮ 1 0 vanishes on the image. (The image is also governed by suitable triangle inequalities.)
We will see that V 4 , R 2 V_{4,\mathbb{R}^{2}} V 4 , R 2 Theorem 7.1 , that the number of 4-tuples of points in Λ \Lambda Λ [ R , R ] 6 [R,R]^{6} [ R , R ] 6 R ϵ R^{\epsilon} R ϵ
But V 5 , R 3 V_{5,\mathbb{R}^{3}} V 5 , R 3 V 4 , R 2 V_{4,\mathbb{R}^{2}} V 4 , R 2
We will then work with a variety formed by fixing some initial points P 2 , … , P m + 2 P_{2},\ldots,P_{m+2} P 2 , … , P m + 2
R m ⟶ R m + 1 , P 1 ↦ ( d ( P 1 , P 2 ) , … , d ( P 1 , P m + 2 ) ) . \mathbb{R}^{m}\longrightarrow\mathbb{R}^{m+1},\qquad P_{1}\mapsto\left(d(P_{1},P_{2}),\ldots,d(P_{1},P_{m+2})\right). R m ⟶ R m + 1 , P 1 ↦ ( d ( P 1 , P 2 ) , … , d ( P 1 , P m + 2 ) ) . Further, to avoid awkward issues about weakly specials being contained “in full dimension” yet not fully contained in varieties under consideration (as above), it is convenient to work with complex algebraic varieties.
Thus, given P 2 , … , P m + 2 ∈ C m P_{2},\ldots,P_{m+2}\in\mathbb{C}^{m} P 2 , … , P m + 2 ∈ C m V P ⊂ C m + 1 V_{P}\subset\mathbb{C}^{m+1} V P ⊂ C m + 1
CM m ( d ˉ ) = 0. \operatorname{CM}_{m}(\bar{d})=0. CM m ( d ˉ ) = 0. In what follows, by points in R M \mathbb{R}^{M} R M affine general position we mean that no m + 1 m+1 m + 1 m − 1 m-1 m − 1
Proposition
Proposition. If the m + 1 m+1 m + 1 P 2 , … , P m + 2 ∈ R m P_{2},\ldots,P_{m+2}\in\mathbb{R}^{m} P 2 , … , P m + 2 ∈ R m affine general position then any weakly special subvariety of positive dimension contained in V P V_{P} V P
Proof. This can be seen in the structure of CM m ( d ˉ ) \operatorname{CM}_{m}(\bar{d}) CM m ( d ˉ ) d 1 i d_{1i} d 1 i d i 1 d_{i1} d i 1 CM m ( d ˉ ) \operatorname{CM}_{m}(\bar{d}) CM m ( d ˉ ) [7 Proposition 9.7.3.2] (the determinant, suitably normalized, is equal to the squared volume of the polytope formed by the points) and the affine general position. □
We are now ready to prove Theorem 7.1.
Proof of Theorem 7.1
The proof is by induction on m m m
For m = 1 m=1 m = 1 Λ ⊂ R \Lambda\subset\mathbb{R} Λ ⊂ R κ ∈ R + , κ ≠ ± 1 / 2 \kappa\in\mathbb{R}_{+},\kappa\ne\pm1/2 κ ∈ R + , κ = ± 1/2 Λ \Lambda Λ P 1 , P 2 P_{1},P_{2} P 1 , P 2 P 1 = 0 P_{1}=0 P 1 = 0 P 2 = R 0 > 0 P_{2}=R_{0}>0 P 2 = R 0 > 0 Q Q Q ( d ( Q , P 1 ) , d ( Q , P 2 ) ) ∈ R 2 \left(d(Q,P_{1}),d(Q,P_{2})\right)\in\mathbb{R}^{2} ( d ( Q , P 1 ) , d ( Q , P 2 ) ) ∈ R 2 V P ⊂ C 2 V_{P}\subset\mathbb{C}^{2} V P ⊂ C 2 CM 1 \operatorname{CM}_{1} CM 1 # MSV ( W , T ) \#\operatorname{MSV}(W,T) # MSV ( W , T ) W W W T T T
# MSV ( V P ‾ , T ) ≤ c ( ϵ ) T ϵ \#\operatorname{MSV}(V_{\overline{P}},T) \leq c(\epsilon)T^{\epsilon} # MSV ( V P , T ) ≤ c ( ϵ ) T ϵ for this family of varieties (i.e. uniformly in P 0 P_0 P 0 P 1 P_1 P 1
Now suppose the conclusion holds for all m ′ < m m' < m m ′ < m Λ ⊂ R m \Lambda\subset\mathbb{R}^m Λ ⊂ R m Λ \Lambda Λ m + 1 m+1 m + 1 P ‾ = ( P 0 , … , P m ) \overline{P}=(P_0,\ldots,P_m) P = ( P 0 , … , P m ) V P ‾ V_{\overline{P}} V P Proposition 7.2 , and a positive-dimensional weakly special subvariety of V P ‾ V_{\overline{P}} V P Q Q Q P i ≠ P j P_i \neq P_j P i = P j ( m + 1 ) m / 2 (m+1)m/2 ( m + 1 ) m /2 Q Q Q
If Λ \Lambda Λ m + 1 m+1 m + 1 Λ \Lambda Λ □ \square □
As noted, V 5 , R 3 V_{5,\mathbb{R}^3} V 5 , R 3
However, in R 2 \mathbb{R}^2 R 2 R 2 \mathbb{R}^2 R 2 T ϵ T^\epsilon T ϵ T ϵ T^\epsilon T ϵ ϵ \epsilon ϵ
In the following, V 4 , C 2 V_{4,\mathbb{C}^2} V 4 , C 2 C 6 \mathbb{C}^6 C 6 CM 2 \operatorname{CM}_2 CM 2
Proposition 7.4. The distance variety V 4 , C 2 ⊂ C 6 V_{4,\mathbb{C}^2} \subset\mathbb{C}^6 V 4 , C 2 ⊂ C 6
Proof. Very elementary computations will be suppressed.
If V V V T T T T T T
First we see that not all coordinates can be non-constant and equal. This follows from the computation in § 11 below.
So some distance is fixed. Without loss of generality it is d ( P 1 , P 2 ) d(P_1,P_2) d ( P 1 , P 2 ) P 1 = ( 0 , 0 ) P_1=(0,0) P 1 = ( 0 , 0 ) P 2 = ( x 2 , 0 ) P_2=(x_2,0) P 2 = ( x 2 , 0 ) P 3 , P 4 P_3, P_4 P 3 , P 4 L = { ( x 2 / 2 , y ) : y ∈ C } L=\{(x_2/2,y):y\in\mathbb{C}\} L = {( x 2 /2 , y ) : y ∈ C } d 2 d^2 d 2 d ( P 3 , P 1 ) 2 = x 3 2 + y 3 2 = ( x 2 / 2 ) 2 + y 3 2 = d 2 d(P_3,P_1)^2=x_3^2+y_3^2=(x_2/2)^2+y_3^2=d^2 d ( P 3 , P 1 ) 2 = x 3 2 + y 3 2 = ( x 2 /2 ) 2 + y 3 2 = d 2 P 3 ∈ L P_3\in L P 3 ∈ L P 4 P_4 P 4 d ( P 3 , P 4 ) 2 ≠ d 2 d(P_3,P_4)^2\ne d^2 d ( P 3 , P 4 ) 2 = d 2 d 2 d^2 d 2
So there must be another constant distance. Either it is a distance of P 3 P_3 P 3 P 4 P_4 P 4 P 1 P_1 P 1 P 2 P_2 P 2 d ( P 3 , P 1 ) d(P_3,P_1) d ( P 3 , P 1 ) d ( P 3 , P 4 ) d(P_3,P_4) d ( P 3 , P 4 )
In this second case, if the non-constant distances are taken equal, P 3 , P 4 P_3,P_4 P 3 , P 4 L L L d 2 d^2 d 2 P 1 , P 2 P_1,P_2 P 1 , P 2 d 2 d^2 d 2
We consider the first case. Now P 3 P_3 P 3 d 3 d_3 d 3 P 1 P_1 P 1 d 3 2 = x 3 2 + y 3 2 d_3^2=x_3^2+y_3^2 d 3 2 = x 3 2 + y 3 2 P 4 P_4 P 4 P 1 , P 2 , P 3 P_1,P_2,P_3 P 1 , P 2 , P 3 P 4 ∈ L P_4\in L P 4 ∈ L d 2 d^2 d 2 P 4 = ( x 2 / 2 , y 4 ) P_4=(x_2/2,y_4) P 4 = ( x 2 /2 , y 4 ) y 4 2 = d 2 − x 2 2 / 4 y_4^2=d^2-x_2^2/4 y 4 2 = d 2 − x 2 2 /4
d 2 = d 2 ( P 4 , P 3 ) = ( x 2 / 2 − x 3 ) 2 + ( y 4 − y 3 ) 2 d^2=d^2(P_4,P_3)=(x_2/2-x_3)^2+(y_4-y_3)^2 d 2 = d 2 ( P 4 , P 3 ) = ( x 2 /2 − x 3 ) 2 + ( y 4 − y 3 ) 2 and x 3 2 + y 3 2 = d 3 2 x_3^2+y_3^2=d_3^2 x 3 2 + y 3 2 = d 3 2 d ( P 3 , P 2 ) 2 = d 2 d(P_3,P_2)^2=d^2 d ( P 3 , P 2 ) 2 = d 2 P 3 P_3 P 3
So either d ( P 3 , P 2 ) d(P_3,P_2) d ( P 3 , P 2 ) P 4 P_4 P 4 d ( P 4 , P 1 ) d(P_4,P_1) d ( P 4 , P 1 )
Transcendence conjectures and Fuglede’s conjecture
Schanuel’s conjecture and non-generic Bessel zeros
Transcendence properties of values of the exponential function are governed by Schanuel’s conjecture.
Schanuel’s conjecture
Schanuel’s conjecture. ([21 p30] ) Let y 1 , … , y m ∈ C y_1,\ldots,y_m\in\mathbb{C} y 1 , … , y m ∈ C
td Q Q ( y 1 , … , y m , e y 1 , … , e y m ) ≥ m \operatorname{td}_{\mathbb{Q}}\mathbb{Q}(y_1,\ldots,y_m,e^{y_1},\ldots,e^{y_m})\ge m td Q Q ( y 1 , … , y m , e y 1 , … , e y m ) ≥ m unless y 1 , … , y m y_1,\ldots,y_m y 1 , … , y m Q \mathbb{Q} Q
We show that, for odd m ≠ ± 1 m\ne\pm1 m = ± 1 J m / 2 J_{m/2} J m /2 Q \mathbb{Q} Q R m \mathbb{R}^m R m m ≥ 3 m\ge3 m ≥ 3 J ± 1 / 2 J_{\pm1/2} J ± 1/2
Proposition. Assume Schanuel’s conjecture. Let m ≥ 1 m\ge1 m ≥ 1 J m + 1 / 2 J_{m+1/2} J m + 1/2 Q \mathbb{Q} Q
Proof. Let m ≥ 1 m \ge1 m ≥ 1 x x x J m + 1 / 2 ( x ) J_{m+1/2}(x) J m + 1/2 ( x )
− 1 = − R ( x ) e 2 i x -1=-R(x)e^{2ix} − 1 = − R ( x ) e 2 i x where R R R Lemma 5.2 . We know that R R R R ( x ) R ( − x ) = 1 R(x)R(-x)=1 R ( x ) R ( − x ) = 1 y = 2 i x y=2ix y = 2 i x S ( y ) = R ( y / 2 i ) S(y)=R(y/2i) S ( y ) = R ( y /2 i ) y y y S ( y ) = e y S(y)=e^y S ( y ) = e y S ( y ) S ( − y ) = 1 S(y)S(-y)=1 S ( y ) S ( − y ) = 1 y i y_i y i y i = ± y j y_i=\pm y_j y i = ± y j
Schanuel’s conjecture immediately implies that any non-zero root y i y_i y i
Now we suppose that any n n n y i y_i y i i = 1 , … , n i=1,\ldots,n i = 1 , … , n y i ≠ ± y j y_i\ne\pm y_j y i = ± y j i ≠ j i\ne j i = j Q \mathbb{Q} Q n + 1 n+1 n + 1 y n + 1 y_{n+1} y n + 1
td ( y 1 , … , y n + 1 , e y 1 , … , e y n + 1 ) ≥ n + 1 \operatorname{td}\left(y_1,\ldots,y_{n+1},e^{y_1},\ldots,e^{y_{n+1}}\right)\ge n+1 td ( y 1 , … , y n + 1 , e y 1 , … , e y n + 1 ) ≥ n + 1 unless y 1 , … , y n + 1 y_1,\ldots,y_{n+1} y 1 , … , y n + 1 Q \mathbb{Q} Q e y i e^{y_i} e y i y i y_i y i
td ( y 1 , … , y n + 1 ) ≥ n + 1 \operatorname{td}\left(y_1,\ldots,y_{n+1}\right)\ge n+1 td ( y 1 , … , y n + 1 ) ≥ n + 1 with the same possible exception. Such a Q \mathbb{Q} Q
y n + 1 = ∑ j = 1 n q j y j y_{n+1}=\sum_{j=1}^{n}q_jy_j y n + 1 = j = 1 ∑ n q j y j for q j ∈ Q q_j\in\mathbb{Q} q j ∈ Q y 1 , … , y n y_1,\ldots,y_n y 1 , … , y n q j = ℓ j / d q_j=\ell_j/d q j = ℓ j / d d d d ℓ j \ell_j ℓ j
S ( ∑ j = 1 n q j y j ) d = S ( y n + 1 ) d = e d y n + 1 = ∏ j = 1 n e d q j y j = ∏ j = 1 n S ( y j ) ℓ j . S\left(\sum_{j=1}^{n}q_jy_j\right)^d=S(y_{n+1})^d=e^{d y_{n+1}}=\prod_{j=1}^{n}e^{d q_j y_j}=\prod_{j=1}^{n}S(y_j)^{\ell_j}. S ( j = 1 ∑ n q j y j ) d = S ( y n + 1 ) d = e d y n + 1 = j = 1 ∏ n e d q j y j = j = 1 ∏ n S ( y j ) ℓ j . As the y j y_j y j q j q_j q j q j q_j q j
S ( q y ) d = S ( y ) ℓ S(qy)^d=S(y)^\ell S ( q y ) d = S ( y ) ℓ which is impossible, considering degrees (as S S S ℓ = ± d \ell=\pm d ℓ = ± d q = ± 1 q=\pm1 q = ± 1 y n + 1 = ± y j y_{n+1}=\pm y_j y n + 1 = ± y j
Concerning the algebraic independence of roots of general non-trivial polynomial-exponential equations and connections with Zilber’s existential closure conjecture see [23 , 24 ] .
Algebraic independence of zeros and Fuglede’s conjecture
Let m ≥ 1 m \ge1 m ≥ 1 M = ( m + 2 ) ( m + 1 ) / 2 M = (m+2)(m+1)/2 M = ( m + 2 ) ( m + 1 ) /2 m + 2 m+2 m + 2 R m \mathbb{R}^m R m
d ˉ = ( d i j ) = ( d ( P i , P j ) , i ≠ j ) ∈ R M . \bar{d}=(d_{ij})=(d(P_i,P_j),i\ne j)\in\mathbb{R}^{M}. d ˉ = ( d ij ) = ( d ( P i , P j ) , i = j ) ∈ R M . Set ℓ = m + 2 \ell=m+2 ℓ = m + 2 d ˉ ∈ R M \bar{d}\in\mathbb{R}^{M} d ˉ ∈ R M P 1 , … , P m + 2 ∈ R m P_1,\ldots,P_{m+2}\in\mathbb{R}^m P 1 , … , P m + 2 ∈ R m CM m ( d ˉ ) \operatorname{CM}_m(\bar{d}) CM m ( d ˉ )
Proposition. Let m ≥ 1 m \ge1 m ≥ 1 M = ( m + 2 ) ( m + 1 ) / 2 M=(m+2)(m+1)/2 M = ( m + 2 ) ( m + 1 ) /2 x i , i = 1 , … , M x_i,i=1,\ldots,M x i , i = 1 , … , M Q \mathbb{Q} Q m + 2 m+2 m + 2 R m \mathbb{R}^m R m x i x_i x i
Proof. Without allowing repetitions the conclusion is clear as P = CM m ( d ˉ ) P=\operatorname{CM}_m(\bar{d}) P = CM m ( d ˉ ) Q \mathbb{Q} Q P = X − Y P=X-Y P = X − Y P ( a , b ) ≠ 0 P(a,b)\ne0 P ( a , b ) = 0 a , b a,b a , b P ( a , a ) = 0 P(a,a)=0 P ( a , a ) = 0 P P P □ \square □
We need to show that, for an indeterminate T T T ( ℓ + 1 ) × ( ℓ + 1 ) (\ell+1)\times(\ell+1) ( ℓ + 1 ) × ( ℓ + 1 ) CM m ( T , … , T ) \operatorname{CM}_m(T,\ldots,T) CM m ( T , … , T ) Z [ T ] \mathbb{Z}[T] Z [ T ] CM m ( T ˉ ) = \operatorname{CM}_m(\bar{T})= CM m ( T ˉ ) =
det ( 0 T 2 T 2 ⋯ T 2 1 T 2 0 T 2 ⋯ T 2 1 T 2 T 2 0 ⋯ T 2 1 ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ T 2 T 2 T 2 ⋯ 0 1 1 1 1 1 1 0 ) = det ( 0 T 2 T 2 ⋯ T 2 1 T 2 − T 2 0 ⋯ 0 0 T 2 0 − T 2 ⋯ 0 0 ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ T 2 0 0 ⋯ − T 2 0 1 1 1 1 1 0 ) \det\begin{pmatrix}
0&T^2&T^2&\cdots&T^2&1\\
T^2&0&T^2&\cdots&T^2&1\\
T^2&T^2&0&\cdots&T^2&1\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\
T^2&T^2&T^2&\cdots&0&1\\
1&1&1&1&1&0
\end{pmatrix}
=
\det\begin{pmatrix}
0&T^2&T^2&\cdots&T^2&1\\
T^2&-T^2&0&\cdots&0&0\\
T^2&0&-T^2&\cdots&0&0\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\
T^2&0&0&\cdots&-T^2&0\\
1&1&1&1&1&0
\end{pmatrix} det 0 T 2 T 2 ⋮ T 2 1 T 2 0 T 2 ⋮ T 2 1 T 2 T 2 0 ⋮ T 2 1 ⋯ ⋯ ⋯ ⋱ ⋯ 1 T 2 T 2 T 2 ⋮ 0 1 1 1 1 ⋮ 1 0 = det 0 T 2 T 2 ⋮ T 2 1 T 2 − T 2 0 ⋮ 0 1 T 2 0 − T 2 ⋮ 0 1 ⋯ ⋯ ⋯ ⋱ ⋯ 1 T 2 0 0 ⋮ − T 2 1 1 0 0 ⋮ 0 0 where the second expression is obtained by subtracting the first row from rows 2 2 2 ℓ \ell ℓ ℓ × ℓ \ell\times\ell ℓ × ℓ
= ϵ det ( T 2 − T 2 ⋯ 0 T 2 0 ⋯ 0 ⋮ ⋮ ⋱ ⋮ T 2 0 ⋯ − T 2 1 1 1 1 ) = ϵ det ( T 2 − 2 T 2 ⋯ − T 2 T 2 − T 2 ⋯ − T 2 ⋮ ⋮ ⋱ ⋮ T 2 − T 2 ⋯ − 2 T 2 1 0 0 0 ) =\epsilon\det\begin{pmatrix}
T^2&-T^2&\cdots&0\\
T^2&0&\cdots&0\\
\vdots&\vdots&\ddots&\vdots\\
T^2&0&\cdots&-T^2\\
1&1&1&1
\end{pmatrix}
=\epsilon\det\begin{pmatrix}
T^2&-2T^2&\cdots&-T^2\\
T^2&-T^2&\cdots&-T^2\\
\vdots&\vdots&\ddots&\vdots\\
T^2&-T^2&\cdots&-2T^2\\
1&0&0&0
\end{pmatrix} = ϵ det T 2 T 2 ⋮ T 2 1 − T 2 0 ⋮ 0 1 ⋯ ⋯ ⋱ ⋯ 1 0 0 ⋮ − T 2 1 = ϵ det T 2 T 2 ⋮ T 2 1 − 2 T 2 − T 2 ⋮ − T 2 0 ⋯ ⋯ ⋱ ⋯ 0 − T 2 − T 2 ⋮ − 2 T 2 0 where ϵ = ( − 1 ) ℓ \epsilon= (-1)^{\ell} ϵ = ( − 1 ) ℓ ( ℓ − 1 ) × ( ℓ − 1 ) (\ell- 1) \times(\ell- 1) ( ℓ − 1 ) × ( ℓ − 1 )
− det ( − 2 T 2 − T 2 ⋯ − T 2 − T 2 − 2 T 2 ⋯ − T 2 ⋮ ⋮ ⋱ ⋮ − T 2 − T 2 ⋯ − 2 T 2 ) = ϵ T 2 ( ℓ − 1 ) det ( 2 1 ⋯ 1 1 2 ⋯ 1 ⋮ ⋮ ⋱ ⋮ 1 1 ⋯ 2 ) = ( − 1 ) ℓ ℓ T 2 ( ℓ − 1 ) \begin{aligned}
-\det\begin{pmatrix}
-2T^2 & -T^2 & \cdots& -T^2 \\
-T^2 & -2T^2 & \cdots& -T^2 \\
\vdots& \vdots& \ddots& \vdots\\
-T^2 & -T^2 & \cdots& -2T^2
\end{pmatrix}
= \epsilon T^{2(\ell-1)}\det\begin{pmatrix}
2 & 1 & \cdots& 1 \\
1 & 2 & \cdots& 1 \\
\vdots& \vdots& \ddots& \vdots\\
1 & 1 & \cdots& 2
\end{pmatrix} \\
&= (-1)^{\ell}\ell T^{2(\ell-1)}
\end{aligned} − det − 2 T 2 − T 2 ⋮ − T 2 − T 2 − 2 T 2 ⋮ − T 2 ⋯ ⋯ ⋱ ⋯ − T 2 − T 2 ⋮ − 2 T 2 = ϵ T 2 ( ℓ − 1 ) det 2 1 ⋮ 1 1 2 ⋮ 1 ⋯ ⋯ ⋱ ⋯ 1 1 ⋮ 2 = ( − 1 ) ℓ ℓ T 2 ( ℓ − 1 ) as the last displayed matrix has eigenvalues ℓ , 1 , … , 1 \ell,1,\ldots,1 ℓ , 1 , … , 1
For even m m m m / 2 m/2 m /2 J m J_m J m Q \mathbb{Q} Q
A Bessel-Schanuel conjecture
There does not seem to be a Schanuel-type conjecture for the Bessel function stated in the literature. Motivated by our proof of Ax-Schanuel, we offer a formulation here.
In framing our conjecture, we have as “evidence” just our Bessel-Ax-Schanuel and the Siegel-Shidlovskii results [32 , 33 ] and [31 Ch. 9, §7] . These give the analogue of Lindemann-Weierstrass for suitably normalized Bessel functions of the first kind of generic rational non-integral order. For integral order, Y m Y_m Y m E E E
In addition to the Lindemann-Weierstrass theorem, several other theorems and conjectures might have served as guidance to Schanuel in formulating his conjecture. These include the Gelfond-Schneider theorem and Gelfond’s theorem (independence of α β \alpha^{\beta} α β α β 2 \alpha^{\beta^2} α β 2 α ≠ 0 , 1 \alpha\ne0,1 α = 0 , 1 β \beta β [30 Problem 7] to α β i \alpha^{\beta_i} α β i α ≠ 0 , 1 \alpha\ne0,1 α = 0 , 1 Q \mathbb{Q} Q β i \beta_i β i [16 ] and conjecture [30 5.12] (and see also [34 ] ), and the widely expected algebraic independence of logarithms of multiplicatively independent algebraic numbers (see e.g. Lang [21 p31] ).
Schanuel did not have the Ax-Schanuel theorem for the exponential function as a guide: indeed he conjectured it (see [3 ] ), though there were earlier partial results of Skolem and Chabauty (see again [3 ] ).
In addition to Bessel-Ax-Schanuel and Siegel-Shidlovskii, our formulation is in keeping with a conjecture of André [1 Conjecture 4.5.1] on the values of holonomic arithmetic Nilsson Gevrey series [1 3.3.1] .
For κ ∈ Q \kappa\in\mathbb{Q} κ ∈ Q
K κ ( x ) = Γ ( κ + 1 ) ( x 2 ) − κ J κ ( x ) K_{\kappa}(x)=\Gamma(\kappa+1)\left(\frac{x}{2}\right)^{-\kappa}J_{\kappa}(x) K κ ( x ) = Γ ( κ + 1 ) ( 2 x ) − κ J κ ( x ) is an E-function in the sense of (and as observed by) Siegel [32 ] . (For a negative integer κ ≤ − 1 \kappa\leq-1 κ ≤ − 1 κ \kappa κ
J κ ∗ ( x ) = Γ ( κ + 1 ) J κ ( x ) . J_{\kappa}^{*}(x)=\Gamma(\kappa+1)J_{\kappa}(x). J κ ∗ ( x ) = Γ ( κ + 1 ) J κ ( x ) . The rational power of x x x J κ ∗ ( x ) J_{\kappa}^{*}(x) J κ ∗ ( x ) κ \kappa κ
For non-negative integer κ \kappa κ J κ ∗ = J κ J_{\kappa}^{*}=J_{\kappa} J κ ∗ = J κ
Then for non-integral rational κ \kappa κ W κ ∗ W_{\kappa}^{*} W κ ∗ J κ ∗ J_{\kappa}^{*} J κ ∗ Y κ ∗ = J − κ ∗ Y_{\kappa}^{*}=J_{-\kappa}^{*} Y κ ∗ = J − κ ∗ J κ ∗ J_{\kappa}^{*} J κ ∗ J − κ ∗ J_{-\kappa}^{*} J − κ ∗ Q \mathbb{Q} Q J κ ∗ J_{\kappa}^{*} J κ ∗
Wr ( J κ ∗ , Y κ ∗ ) = − 2 κ x . \operatorname{Wr}(J_{\kappa}^{*},Y_{\kappa}^{*})=-\frac{2\kappa}{x}. Wr ( J κ ∗ , Y κ ∗ ) = − x 2 κ . For integer order k k k k k k Y k Y_k Y k k = 0 k=0 k = 0 [5 7.24, eqn. (33)]
π Y 0 ( x ) = 2 J 0 ( x ) ( log ( x / 2 ) + γ ) − 2 ∑ j = 0 ∞ ( − 1 ) j j ! j ! ( x 2 ) 2 j H j \pi Y_0(x)=2J_0(x)\left(\log(x/2)+\gamma\right)-2\sum_{j=0}^{\infty}\frac{(-1)^j}{j!j!}\left(\frac{x}{2}\right)^{2j}H_j π Y 0 ( x ) = 2 J 0 ( x ) ( log ( x /2 ) + γ ) − 2 j = 0 ∑ ∞ j ! j ! ( − 1 ) j ( 2 x ) 2 j H j where H j = 1 + 1 / 2 + … 1 / j H_j=1+1/2+\ldots1/j H j = 1 + 1/2 + … 1/ j H 0 = 0 H_0=0 H 0 = 0 γ \gamma γ
Y 0 ∗ ( x ) = J 0 ( x ) log x + A 0 ( x ) , A 0 ( x ) = − ∑ j = 0 ∞ ( − 1 ) j j ! j ! ( x 2 ) 2 j H j . Y_0^{*}(x)=J_0(x)\log x+A_0(x),\qquad A_0(x)=-\sum_{j=0}^{\infty}\frac{(-1)^j}{j!j!}\left(\frac{x}{2}\right)^{2j}H_j. Y 0 ∗ ( x ) = J 0 ( x ) log x + A 0 ( x ) , A 0 ( x ) = − j = 0 ∑ ∞ j ! j ! ( − 1 ) j ( 2 x ) 2 j H j . We note that A 0 A_0 A 0 Y 0 Y_0 Y 0 [1 ] .
For other (positive) integer orders we proceed similarly. We have
π Y n ( x ) = 2 ( γ + log ( x / 2 ) ) J n ( x ) − ∑ m = 0 n − 1 ( n − m − 1 ) ! m ! ( x 2 ) 2 m − n − ∑ m = 0 ∞ ( − 1 ) m ( H m + n + H m ) m ! ( n + m ) ! ( x 2 ) n + 2 m . \pi Y_n(x)=2\left(\gamma+\log(x/2)\right)J_n(x)-\sum_{m=0}^{n-1}\frac{(n-m-1)!}{m!}\left(\frac{x}{2}\right)^{2m-n}-\sum_{m=0}^{\infty}\frac{(-1)^m(H_{m+n}+H_m)}{m!(n+m)!}\left(\frac{x}{2}\right)^{n+2m}. π Y n ( x ) = 2 ( γ + log ( x /2 ) ) J n ( x ) − m = 0 ∑ n − 1 m ! ( n − m − 1 )! ( 2 x ) 2 m − n − m = 0 ∑ ∞ m ! ( n + m )! ( − 1 ) m ( H m + n + H m ) ( 2 x ) n + 2 m . (see e.g. [5 7.24, eqn. (32)] ). So we set
Y n ∗ ( x ) = ( π / 2 ) Y n ( x ) − ( γ − log 2 ) J n ( x ) . Y_n^{*}(x)=(\pi/2)Y_n(x)-(\gamma-\log2)J_n(x). Y n ∗ ( x ) = ( π /2 ) Y n ( x ) − ( γ − log 2 ) J n ( x ) . which is (but for the Laurent polynomial) a holonomic arithmetic Nilsson Gevrey series. Then, for non-negative integer κ \kappa κ
Wr ( J κ ∗ , Y κ ∗ ) = 1 / x . \operatorname{Wr}(J_{\kappa}^{*},Y_{\kappa}^{*})=1/x. Wr ( J κ ∗ , Y κ ∗ ) = 1/ x . For a negative integer κ \kappa κ κ 2 \kappa^2 κ 2
J κ ∗ = J κ = ( − 1 ) κ J − κ , Y κ ∗ = ( − 1 ) κ Y − κ . J_\kappa^* = J_\kappa= (-1)^\kappa J_{-\kappa}, \qquad Y_\kappa^* = (-1)^\kappa Y_{-\kappa}. J κ ∗ = J κ = ( − 1 ) κ J − κ , Y κ ∗ = ( − 1 ) κ Y − κ . For κ ∈ Q \kappa\in\mathbb{Q} κ ∈ Q W κ ∗ W_\kappa^* W κ ∗ J κ ∗ , Y κ ∗ J_\kappa^*,Y_\kappa^* J κ ∗ , Y κ ∗ x x x Q ‾ \overline{\mathbb{Q}} Q Q ( x , W κ ∗ ( x ) ) \mathbb{Q}(x,W_\kappa^*(x)) Q ( x , W κ ∗ ( x )) Q ( x ) \mathbb{Q}(x) Q ( x )
Conjecture 8.4. Let κ ∈ Q \ ( Z + 1 / 2 ) \kappa\in\mathbb{Q}\backslash(\mathbb{Z}+1/2) κ ∈ Q \ ( Z + 1/2 ) y 1 , … , y m ∈ C y_1,\ldots,y_m \in\mathbb{C} y 1 , … , y m ∈ C x 1 , … , x n ∈ C × x_1,\ldots,x_n \in\mathbb{C}^{\times} x 1 , … , x n ∈ C ×
td Q ( y 1 , … , y m , e y 1 , … , e y m , x 1 , … , x n , W κ ∗ ( x 1 ) , … , W κ ∗ ( x n ) ) ≥ m + 3 n , \operatorname{td}_{\mathbb{Q}}\left(y_1,\ldots,y_m,e^{y_1},\ldots,e^{y_m},x_1,\ldots,x_n,W_\kappa^*(x_1),\ldots,W_\kappa^*(x_n)\right) \ge m+3n, td Q ( y 1 , … , y m , e y 1 , … , e y m , x 1 , … , x n , W κ ∗ ( x 1 ) , … , W κ ∗ ( x n ) ) ≥ m + 3 n , unless either
(A) y 1 , … , y m y_1,\ldots,y_m y 1 , … , y m Q \mathbb{Q} Q
(B) x i = ± x j x_i=\pm x_j x i = ± x j i ≠ j i\ne j i = j
Note that, while we normalized to remove π \pi π Y n Y_n Y n n = 0 n=0 n = 0 Q ( π ) \mathbb{Q}(\pi) Q ( π ) 3 n 3n 3 n x 1 , … , x n x_1,\ldots,x_n x 1 , … , x n ± \pm ± E E E [19 ] . For a related conjecture (with a related provenance) see [22 ] .
One can make a corresponding formulation involving normalized Bessel functions of several different generic rational orders κ 1 , … , κ ℓ \kappa_1,\ldots,\kappa_\ell κ 1 , … , κ ℓ κ i ± κ j ∉ Z \kappa_i\pm\kappa_j\notin\mathbb{Z} κ i ± κ j ∈ / Z i ≠ j i\ne j i = j y j ∈ C , x i ∈ C × y_j\in\mathbb{C},x_i\in\mathbb{C}^{\times} y j ∈ C , x i ∈ C × m + 3 n ℓ m+3n\ell m + 3 n ℓ
Dept. of Mathematics, Weizmann Institute of Science, Rehovot, Israel
and (GB and JP)
School of Mathematics, Institute for Advanced Study, Princeton, USA
gal.binyamini@weizmann.ac.il
avner-ephraiem.kiro@weizmann.ac.il
gady.kozma@weizmann.ac.il
jonathan.pila@maths.ox.ac.uk
