Novel properties of polynomials

Introduction and statement of the results

The main purpose of this paper is to establish new divisibility criteria for complex polynomials and explicit relations between their coefficients and subsets of roots. We begin by recalling the symmetric polynomials that are involved in this study.

For $\vec{z}=(z_1,\ldots,z_n)\in\mathbb{C}^n$ and $k\in\mathbb{N}$, the elementary symmetric polynomial $E_k(\vec{z})$ and the complete homogeneous symmetric polynomial $H_k(\vec{z})$ are defined as

We set $E_0(\vec{z})=H_0(\vec{z})=1$ and $E_k(\vec{z})=0$ when $k>n$. Both polynomials are homogeneous of degree $k$ and are invariant by permutation of the variables. We have used [12, 16] for the definitions and basic properties of these polynomials, some of which are recalled in §2.

The classical Vieta formulas express the coefficients of a complex polynomial in terms of the elementary symmetric polynomials evaluated at its roots. Specifically, if

2020 Mathematics Subject Classification. 12D05 05E05 11C08 11A07.

Key words and phrases. Homogeneous symmetric polynomials, the Vieta formulas, roots of unity, Kronecker polynomials.

with $a_j \in\mathbb{C}$ and $a_0 \ne0$, then

Since roots of polynomials should be considered with their multiplicities, the language of multisets provides the right framework for some of our results and proofs.

We associate to a given vector $\vec{y}:=(y_0,y_1,\ldots,y_{n-1})\in\mathbb{C}^n$ the multiset of its components, denoted by

While the order of the components matters in the vector $\vec{y}$, only the values of the elements and their multiplicities are important in the multiset $Y$.

Given $y\in\mathbb{C}$, the multiplicity $\mathrm{m}(y;Y)$ is the number of occurrences of $y$ in $Y$. Plainly, $\mathrm{m}(y;Y)=0$ if and only if $y$ does not occur in $Y$.

The support of $Y$ is the set of its distinct elements. It is denoted as $\operatorname{supp}Y$. The cardinality of $Y$ is $|Y|:=\sum_{y\in\operatorname{supp}Y}\mathrm{m}(y;Y)=n$.

Given multisets $X$, $Y$, their multiset union, denoted by $X\sqcup Y$, consists of all $y$ with multiplicity $\mathrm{m}(y;X\sqcup Y):=\mathrm{m}(y;X)+\mathrm{m}(y;Y)\ge1$.

We say that $X$ is a submultiset of $Y$, and write $X\sqsubseteq Y$, if $\operatorname{supp}X\subseteq\operatorname{supp}Y$ and $\mathrm{m}(y;X)\le\mathrm{m}(y;Y)$ for every $y\in\operatorname{supp}X$.

If $X\sqsubseteq Y$, then the multiset complement of $X$ in $Y$, denoted by $\widetilde{X}$, is formed with all $y\in\operatorname{supp}Y$ with multiplicity $\mathrm{m}(y;\widetilde{X}):=\mathrm{m}(y;Y)-\mathrm{m}(y;X)$.

Note that the supports of $X$ and $\widetilde{X}$ may overlap. For instance, given the multisets $Y:=\{a,a,b,b,c\}$ and $X:=\{a,a,b\}$, one has $X\sqsubseteq Y$, $\operatorname{supp}Y=\{a,b,c\}$, and $\widetilde{X}=\{b,c\}$.

Since symmetric polynomials are invariant under permutation of their variables, we will indifferently use the notation $\mathcal{E}_k(Y)$ and $\mathcal{E}_k(\vec{y})$ (resp. $\mathcal{H}_k(Y)$ and $\mathcal{H}_k(\vec{y})$). We also adopt the convention

Main results

As already mentioned, when counted with their multiplicity, the roots of a complex polynomial form a multiset, that we call the roots multiset. In this subsection we state explicit relations between the coefficients of a complex polynomial when a submultiset of its roots multiset is given. Further, we characterize such submultisets.

whose roots multiset is $J_n$; define

The following properties are equivalent:

(a) $J_n \subseteq X_m$.

(b) $Q_n(z)$ divides $P(z)$, and $P(z) = Q_n(z) \sum_{k=0}^{m-n} c_k z^{m-n-k}$.

(c) $c_k = 0$ for every $k \in\{m-n+1,m-n+2,\ldots,m\}$.

This theorem considerably improves on [5 Th. 1.1], where we have just established that the property (b) implies (c), whereas here we show that the converse holds.

Further, the Vieta formulas (1.1) are recovered by taking $J_n = X_m$, i.e., $n=m$, in Theorem 1.1. In order to see it, let us first recall a well known identity relating complex and homogeneous symmetric polynomials given by

See Proposition 2.4 below for a more general identity. Then, note that for $J_n = X_m$ the condition (c) of Theorem 1.1 becomes

In particular, for $k=1$ this gives

that is $\mathcal{H}_1(X_m)=E_1(X_m)=-a_1/a_0$, i.e., the formula (1.1) for $k=1$.

Now, for $k=2$ observe that (1.3) yields $\mathcal{H}_2(X_m)=E_1(X_m)^2-E_2(X_m)$. Thus, from (1.4) for $k=1,2$ it follows that

which is (1.1) for $k=2$. The remaining case $3\le k\le m$ can be accomplished in a completely analogous way.

Before presenting some consequences of Theorem 1.1, let us introduce the following notation. Given $h\in\{1,2,\ldots,m\}$, we group the terms of the complex polynomial $\mathcal{P}(z):=\sum_{k=0}^{m}a_k z^{m-k}$, with $\deg\mathcal{P}(z)=m\in\mathbb{N}$, according to their exponents modulo $h$, so that

where

Hereafter, $[x]$ denotes the integer part of the real number $x$.

Note that

The next corollary characterizes the divisibility of $\mathcal{P}(z)$ by the complex polynomial $\mathcal{Q}(z) = \sum_{j=0}^{n} b_jz^{n-j}$, with $n \le m$, in terms of their coefficients.

Let us first recall that, given $k \in\mathbb{N}_0$ and $\vec{\alpha} := (\alpha_1,\ldots,\alpha_n) \in\mathbb{N}_0^n$, the multinomial coefficient is defined as [15 §1.2]

Further, we introduce the following notation:

Further results

In [6] we have recently proved that if the roots multiset $X_m$ of a complex and monic polynomial $\mathcal{P}(z)$ consists of $m = \deg\mathcal{P}(z)$ distinct $n$th roots of unity, then the cofactor of $\mathcal{P}(z)$ in $z^n-1$ is given by

This result motivates the more general study of the product $\mathcal{P}(z)Q_r(z)$ when $X_m$ is an arbitrary root multiset and

The next proposition identifies $Q_r(z)$ as the quotient in the division of $z^{m+r}$ by $\mathcal{P}(z)$ and gives an explicit formula for the remainder. An equivalent formula appears in [18 Eq. (18)]; we include a short proof of our proposition in §3.

A criterion for Kronecker polynomials

A monic polynomial with integer coefficients is a Kronecker polynomial if all its zeros belong to the closed unit disk. A classical theorem of Kronecker shows that every nonzero root of such a polynomial is in fact a root of unity [10]. Consequently, if $\mathcal{K}(z)$ is a Kronecker polynomial, then it admits a factorization of the form

where $K\subset\mathbb{N}$ is finite, $r\in\mathbb{N}_0$, $b_j\in\mathbb{N}$, and $\Phi_j(z)$ is the cyclotomic polynomial of index $j$. Conversely, every polynomial of the form (1.7) is a Kronecker polynomial.

Recall that the cyclotomic polynomial of index $n \in\mathbb{N}$ is defined as

with $\deg\Phi_n(z)=\varphi(n):=\lvert\{s\in\mathbb{N}:1\le s\le n\text{ and }\gcd(s,n)=1\}\rvert$, the Euler totient function. It is well known that its coefficients belong to $\mathbb{Z}$, it is irreducible over $\mathbb{Q}$, and $z^n-1=\prod_{d\mid n}\Phi_d(z)$. We refer the reader to [8] §13.6 and also to the survey papers [14, 17] for an overview of these polynomials.

The following proposition characterizes Kronecker polynomials whose roots are nonzero. It is used to derive the criterion of Theorem 1.6 below. In §3 we include a proof of such a proposition.

Remark 1.2. Ayache, Echi, and Naimi introduced in [1] the more restrictive class of absolutely Kronecker polynomials characterized by the condition that $\mathcal{P}(z)$ divides $\mathcal{P}(z^t)$ in $\mathbb{C}[z]$ for every $t\in\mathbb{N}$.

Here we can combine Proposition 1.5 with the divisibility criteria given by Corollary 1.3 to obtain the following result, which we prove in §3.

Applications to polynomials with restricted coefficients

Given a polynomial $P(z)$ with integer coefficients and a finite set $D \subset\mathbb{Z}$, a question to ask is whether $P(z)$ divides a nonzero polynomial whose coefficients belong to $D$. Here are some cases of particular interest:

• $D = \{0,1\}$ and constant term 1 (Newman polynomials)

• $D = \{-1,1\}$ (Littlewood polynomials)

• $D = \{-1,0,1\}$ and nonzero constant term (Borwein polynomials)

A major issue is to determine whether such multiples exist and, when they do, to find their smallest possible degree. These questions have been studied in connection with polynomials of small height, Mahler measure, Pisot and Salem numbers, and finite-state algorithms; see [3], and also [4, 7], the recent [9] and the references therein.

By applying Corollary 1.3 we easily obtain the next divisibility criterion which yields explicit conditions for the existence of multiples of $P(z)$ with coefficients in a prescribed finite set of integers. Indeed, after recalling the definition of $F_j$ given in Theorem 1.6, it suffices to observe that the divisibility of $P(z)$ by the polynomial

with $b_0 := 1$, is equivalent to (1.10) below. Thus, we get immediately the following criterion.

In view of Proposition 2.5, all the equations (10) can be written in terms of the coefficients of $P(z)$.

By taking $D = \{0,1\}$, $\{-1,1\}$, or $\{-1,0,1\}$ in the previous theorem, we obtain a characterization of the existence of Newman, Littlewood, or Borwein multiples, respectively, with the additional requirement $b_N = 1$ in the Newman case and $b_N \ne0$ in the Borwein one.

Plainly, such a characterization consists of a finite system of linear equations with solutions in a preassigned set $D \subset\mathbb{Z}$. Consequently, the minimum-degree problem can be formulated as finding the smallest $N \ge m$ for which (1.10) admits solutions in $D$. This formulation might help in computational investigations of Newman, Littlewood, and Borwein multiples.

It is proved in [7 Proposition 13] that every product of cyclotomic polynomials has a Littlewood multiple. Since a Littlewood polynomial is a Borwein polynomial, the existence of a Borwein multiple of a product of cyclotomic polynomials is not new. However, here we present the following property

This proposition and Proposition 1.5 immediately yield the following

The proofs of the aforementioned results are given in § 4. For such proofs we need some old and new properties of the symmetric polynomials $\mathcal{H}_m$ and $\mathcal{E}_m$ which are collected in the next section.

Old and new identities for symmetric polynomials

Generating functions

For $\vec{z}=(z_1,\ldots,z_n)\in\mathbb{C}^n$, the generating functions of $\mathcal{H}_m$ and $\mathcal{E}_m$ are

and

respectively. We regard (12) as an identity of formal power series.

As before, if $Y_n$ is the multiset of the components of $\vec{y}$, then we write $H(t;Y_n)=H(t;\vec{y})$ and $E(t;Y_n)=E(t;\vec{y})$.

In view of (12) and (13), one has

for every $m\geq0$. Though the first formula differentiation is understood formally, the usual product and chain rules remain valid [16]. Moreover,

The properties of the generating functions given in Propositions 2.1 and 2.2 below are well known, but we provide their short proofs for convenience.

The first property relates these generating functions to a polynomial with prescribed roots multiset.

The next proposition extends the identity (15).

Convolution identities

The identities in the next proposition can be found in [12 Chapter I, §5, Example 25(a), p. 92.]. See also [13 Th. 1.1].

In the proof of Corollary 1.3 we apply the following identities for $\mathcal{H}_m$ and $\mathcal{E}_m$. They follow easily by comparing coefficients of $t^m$ in the formal power series after using (2.1), (2.2), and Proposition 2.2.

Both identities in Proposition 2.4 are special cases of the classical formulas for the symmetric functions of a difference of alphabets (finite multisets of variables). Writing $h_m(X)=\mathcal{H}_m(\vec{x})$ and $e_m(X)=\mathcal{E}_m(\vec{x})$ for the alphabet $X=\{x_1,\ldots,x_k\}$, these formulas read

and Proposition 2.4 follows by replacing $X$ with the alphabet $X\sqcup Y=\{x_1,\ldots,x_k,y_1,\ldots,y_n\}$. For more details, see e.g. [11, 19].

Note also that Proposition 2.4 can be equivalently reformulated in terms of multisets as follows. Let $J\sqsubseteq Y$, with $1\leq|J|\leq|Y|$, and let $\breve{J}$ be the complement multiset of $J$ with respect to $Y$. For every $m\in\mathbb{N}_0$ one has

Applying the latter formula with $J$ and its complement $\breve{J}$ interchanged gives

yielding in particular $\sum_{j=0}^{m}(-1)^j\mathcal{H}_j(J)\mathcal{E}_{m-j}(Y)=0$ if $m>|Y|-|J|$. This reduces to the standard identity (1.3) when $J=Y$.

Expressing $\mathcal{H}_m$ in terms of $\mathcal{E}_k$

In the proof of Corollary 1.3 we apply the following property, which is a special case of a theorem by Gauss (see [2 §2.2.15]).

where

This proposition shows that if the $\mathcal{E}_j(\vec{z}) \in\mathbb{Z}$ for all $j=0,\ldots,n$, then also $\mathcal{H}_m(\vec{z}) \in\mathbb{Z}$ for all $m \in\mathbb{N}_0$.

For the proof of Proposition 2.5 we need the multinomial theorem, which states that [15 §1.2]

for any $x_1,\ldots,x_n \in\mathbb{C}$, $k \in\mathbb{N}_0$. Let us recall also the generalized Leibniz formula

Proof of Proposition 2.5. Clearly, both identities (16) and (17) are trivially true when $m=0$. Thus, we assume that $m \in\mathbb{N}$. By using (12) and (13), we write

where we have let

for brevity. Thus, expanding as a formal power series gives

so that (14) yields $\mathcal{H}_m(\vec{z})=\frac{1}{m!}\left.\sum_{k=0}^{\infty}\frac{\mathrm{d}^m}{\mathrm{d}t^m}F_n(t;\vec{z})^k\right|_{t=0}$.

Now, observe that $F_n^{(0)}(0;\vec{z}):=F_n(0;\vec{z})=0$ and

Therefore, by applying the generalized Leibnitz formula we see that

Since $m_1+\cdots+m_k=m$ can hold with $m_i\in\mathbb{N}$ only when $1\leq k\leq m$, the previous series reduces to a finite sum as

which is (16).

In order to prove (17), first note that the multinomial theorem yields

Since for any $\vec{\alpha}\in\mathbb{N}_0^n$ one has

from (18) we see that

Hence, this reduces to (17) because $\Sigma(\vec{\alpha})\leq w(\vec{\alpha})$ for any $\vec{\alpha}\in\mathbb{N}_0^n$, so that there exists exactly one value $k=\Sigma(\vec{\alpha})\in\{1,\ldots,m\}$ such that $w(\vec{\alpha})=m$. The proposition is proved. □

Remark 2.1. From (18) we obtain the following coefficient formula: if $X_m$ is the roots multiset of the complex polynomial $\mathcal{P}(z)=\sum_{j=0}^{m}a_jz^{m-j}$, then

Indeed, by using the Vieta formulas (1) we see that

where $\mathcal{F}_m(t;X_m)$ is defined as in the proof of Proposition 2.5. Hence, the claimed identity for $\mathcal{H}_n(X_m)$ follows immediately from (18).

Proofs of the results

Proof of Theorem 1.1. Since $\mathcal{P}(z)=a_0\prod_{x\in X_m}(z-x)$, it is plain that (a) and (b) are equivalent. In order to prove the equivalence of (b) and (c), let us set

In view of Proposition 2.1, we can write

where $c_k:=\sum_{s=0}^{\min(k,m)}a_s\mathcal{H}_{k-s}(J_n)$ for every $k\ge0$.

Now, if (b) holds, then $\mathcal{P}(z)=\mathcal{Q}_n(z)S(z)$ for some polynomial $S(z)$ of degree $m-n$, so that $\mathcal{R}(t)=t^{m-n}S(t^{-1})$ is a polynomial of degree at most $m-n$. Consequently, $c_k=0$ for every $k>m-n$, yielding (c).

Conversely, assume that (c) holds true, i.e.,

Since $\mathcal{B}(0)=1$, we can set $\mathcal{B}(t)=1+b_1t+\cdots+b_nt^n$ and compare the coefficients in $\mathcal{B}(t)\mathcal{R}(t)=\mathcal{A}(t)$. For every integer $k>m$ we have that

because the degree of $\mathcal{A}(t)$ is at most $m$. In particular, for $k=m+1$ this is

yielding $c_{m+1}=0$ because $c_m=c_{m-1}=\cdots=c_{m-n+1}=0$ by hypothesis. Thus, we can proceed by induction to show that $c_k=0$ for every $k>m$. Together with the hypothesis (c), this implies that

Since $\frac{P(z)}{Q_n(z)}=z^{m-n}R(z^{-1})$, we can finally write

yielding (b). □

Proof of Corollary 1.2. It is plain that if $G(z)$ divides $P_{h,s}(z)$ for every $s\in\{0,1,\ldots,h-1\}$, then $G(z)$ divides $P(z)$. Conversely, assume that $P(z)=G(z)T(z)$ for some complex polynomial $T(z)$. Thus, we can write

where $T_{h,s}(z)$ is defined in an analogous way as $P_{h,s}(z)$ in (1.5). Hence, the polynomial

is identically zero on $\mathbb{C}$, i.e., all its coefficients are equal to zero. Now, note that $P_{h,s}(z^h)-G(z^h)T_{h,s}(z^h)$ is a polynomial in $z^h$ involving disjoint sets of powers of $z$ for distinct values of $s\in\{0,1,\ldots,h-1\}$. Indeed, it suffices to observe that if $0\leq s,s'<h$ and $j,j'\geq0$, then $s+jh=s'+j'h$ if and only if $s=s'$ and $j=j'$. Thus, being $F(z)$ the zero polynomial, we must conclude that

showing that $G(z)$ divides $P_s(z)$ for every $s=0,\ldots,h-1$, as required. □

Proof of Corollary 1.3. Let $J_n$ be the roots multiset of $Q$, so that $\mathcal{E}_j(J_n)=(-1)^jb_j$ by the Vieta formulas (1.1). Thus, by using Proposition 2.5 we have that

Hence, the claimed formulas follow from Theorem 1.1 for $Q(z)$ divides $P(z)$ if and only if

for every $k\in\{m-n+1,\ldots,m\}$. □

Proof of Proposition 1.4. Set

By using Proposition 2.1 we write

so that $\mathcal{A}(t)S_r(t)=1-\mathcal{A}(t)\sum_{s=r+1}^{\infty}\mathcal{H}_s(X_m)t^s$. Now, after noticing that $\mathcal{A}(t)S_r(t)$ has degree $r+m$, we infer

Since the conditions $s\ge r+1$ and $s+j=k+r$ imply that $j\le k-1$, one has

with

as in (6). Substituting $t=z^{-1}$ and multiplying by $z^{m+r}$, the previous identity gives

as claimed.

Now, let us take $s\in\mathbb{C}$ fixed and write

where plainly $\deg(\mathcal{R}_r+s)<m$. Thus, we see that $\mathcal{P}(z)$ divides $z^{m+r}-s$ if and only if $\mathcal{R}_r(z)=-s$. In view of (6), this is equivalent to saying that $c_{m,r}=-s$ and $c_{k,r}=0$ when $1\le k<m$. □

Proof of Proposition 1.5. (a)⇒(b): suppose that $\mathcal{K}(z)$ is a Kronecker polynomial. Since $\mathcal{K}(0)\ne0$, by Kronecker’s theorem we deduce that all the roots of $\mathcal{K}(z)$ are roots of unity. Assume first that all these roots are simple and let $N$ be a common multiple of their orders. If $t>1$ is such that $t \equiv1 \pmod{N}$, then $\alpha^t=\alpha$ for every root $\alpha$ of $\mathcal{K}(z)$. Thus, we deduce that $\alpha$ is also a root of $\mathcal{K}(z^t)$. Consequently, $\mathcal{K}(z)$ divides $\mathcal{K}(z^t)$ because its roots are simple.

Now, assume that $\alpha_1,\alpha_2,\ldots,\alpha_s$ are the roots of $\mathcal{K}(z)$ which are not simple, let $d_1,d_2,\ldots,d_s\ge2$ be their respective multiplicity. Since all the roots of the polynomial

are roots of unity and simple, by the first part of the proof it follows that $Q(z)$ divides $Q(z^t)$, where $t$ is defined as before. Further, $(z-\alpha_j)^{d_j-1}$ divides $(z^t-\alpha_j)^{d_j-1}$ for $\alpha_j$ is a root of $z^t-\alpha_j$ by definition of $t$. Thus, we conclude that $\mathcal{K}(z)=Q(z)(z-\alpha_1)^{d_1-1}\cdots(z-\alpha_s)^{d_s-1}$ divides $\mathcal{K}(z^t)=Q(z^t)(z^t-\alpha_1)^{d_1-1}\cdots(z^t-\alpha_s)^{d_s-1}$, as claimed.

(b)$\Rightarrow$(a): assume that there exists an integer $t>1$ such that $\mathcal{K}(z)$ divides $\mathcal{K}(z^t)$. This implies that if $\alpha$ is a root of $\mathcal{K}(z)$, then so is $\alpha^t$. Further, it must be $\alpha\ne0$ by the hypothesis $\mathcal{K}(0)\ne0$. Thus, we can iterate this argument and conclude that $\alpha,\alpha^t,\alpha^{t^2},\ldots$ are roots of $\mathcal{K}(z)$. Since there are only finitely many distinct roots, it must be $\alpha^{t^r}=\alpha^{t^s}$ for some distinct integers $r,s\in\mathbb{N}_0$, implying that $\alpha$ is a root of unity. Hence, $\mathcal{K}(z)$ is a Kronecker polynomial.

(a)$\Rightarrow$(c): if $\mathcal{K}(z)$ is a Kronecker polynomial, then it admits a representation as (7), with $r=0$ for $\mathcal{K}(0)\ne0$, namely

where $K\subset\mathbb{N}$ is finite and $b_j\in\mathbb{N}$. Note that if $\alpha$ is a root of $\mathcal{K}(z)$, then $\alpha^j=1$ for some $j\in K$. Therefore, denoting by $N$ the least common multiple of the integers $jb_j$, with $j\in K$, we deduce that all the roots of $\mathcal{K}(z)$ are roots of $z^N-1$.

Since all the roots of any cyclotomic polynomial are simple, we deduce that the multiplicity of any root of $\mathcal{K}(z)$ is not greater than $N$, so that $\mathcal{K}(z)$ must divide $(z^N-1)^s$ for some $s\in\mathbb{N}$.

(c)$\Rightarrow$(a): assume that there exist $s\in\mathbb{N}$ and an integer $N>1$ such that $\mathcal{K}(z)$ divides $(z^N-1)^s$. Thus, if $\alpha$ is a root of $\mathcal{K}(z)$, then it must be $\alpha^N=1$, i.e., $\alpha$ is a root of unity. Hence, $\mathcal{K}(z)$ is a Kronecker polynomial. $\square$

Proof of Theorem 1.6. (a)$\Leftrightarrow$(b): by Proposition 1.5, $\mathcal{K}(z)$ is a Kronecker polynomial if and only if $\mathcal{K}(z)$ divides $\mathcal{K}(z^t)$ for some integer $t>1$. Thus, we can apply Theorem 1.1 by taking $Q_n(z)=\mathcal{K}(z)$ and

with

We deduce that the divisibility of $\mathcal{K}(z^t)$ by $\mathcal{K}(z)$ is equivalent to

for every $k\in\{(t-1)n+1,\ldots,tn\}$, yielding (1.8).

(b)$\Leftrightarrow$(c): first, note that $\mathcal{H}_0(X_n)=F_0=1$. Then, let us apply Proposition 2.5 and (1.1) to see that

for every $k\in\mathbb{N}$. Hence, (1.8) and (1.9) are equivalent. □

Proof of Proposition 1.8. For any given $s\in\{1,\ldots,n\}$ we define

Notice that $\mathcal{P}(\alpha)=0$ if and only if there exists a unique $\ell\in\{1,\ldots,k\}$ such that $\Phi_{j_\ell}(\alpha)=0$. Further, if this is the case, then $d_\ell$ is the multiplicity of $\alpha$ as a root of $\mathcal{P}(z)$ and $\mathcal{P}_s(\alpha)=0$ for every $s\in\{1,\ldots,d_\ell\}$. Thus, $d_\ell$ is also the multiplicity of $\alpha$ as a root of $\mathcal{P}_1(z)\cdots\mathcal{P}_n(z)$, so that it must be

Since for every given $s\in\{1,\ldots,n\}$ one has that $\mathcal{P}_s(z)$ divides $z^{L_s}-1$, where $L_s:=\operatorname{lcm}\{j_\ell:d_\ell\ge s\}$, we choose $N_1=L_1$ and $N_s\in\mathbb{N}$ as a multiple of $L_s$ when $2\le s\le n$ and such that (1.11) is satisfied. Hence, we conclude that $\mathcal{P}(z)$ divides $\mathcal{B}(z)$, as claimed.

In order to see that $\mathcal{B}(z)$ is a Borwein polynomial, first let us write

Then, observe that the inequalities (1.11) ensure that distinct subsets $S$ of $\{1,\ldots,n\}$ give distinct exponents $k_S$. Hence, all coefficients of $\mathcal{B}(z)$ belong to $\{-1,0,1\}$. Moreover, $\mathcal{B}(0)=(-1)^n\ne0$, as required. □

Laura De Carli, Department of Mathematics, Florida International University, Miami, FL 33199, USA.

Email address: [email protected]

Maurizio Laporta, Dipartimento di Matematica e Applicazioni, Università degli Studi di Napoli ”Federico II”, 80126 Napoli, Italy

Email address: [email protected]