Related papers: Linear independence of powers for polynomials

Let a(n,k) be the kth coefficient of the nth cyclotomic polynomial. The first two authors showed in part I that if m is a prime power and n and k range over the non-negative integers, then a(mn,k) assumes every integer value. Here this…

Recently, Davies, Jenssen, Perkins, and Roberts gave a very nice proof of the result (due, in various parts, to Kahn, Galvin-Tetali, and Zhao) that the independence polynomial of a $d$-regular graph is maximized by disjoint copies of…

We consider a space of complex polynomials of degree $n\ge 3$ with $n-1$ distinguished periodic orbits. We prove that the multipliers of these periodic orbits considered as algebraic functions on that space, are algebraically independent…

Let $m\ge 2$ be an integer, $K$ an algebraic number field and $\alpha\in K\setminus \{0,-1\}$ with sufficiently small absolute value. In this article, we provide a new lower bound for linear form in…

In this paper we construct an entire function of two variables having the property that its values and its partial derivatives of any order at any distinct algebraic points are algebraically independent. Such an entire function is generated…

Let A be a finite set of integers. For a polynomial f(x_1,...,x_n) with integer coefficients, let f(A) = {f(a_1,...,a_n) : a_1,...,a_n \in A}. In this paper it is proved that for every pair of normalized binary linear forms f(x,y)=u_1x+v_1y…

The Casas--Alvero conjecture predicts that every univariate polynomial $f$ over a field $K$ of characteristic zero having a common factor with each of its derivatives $H\_i(f)$ is a power of a linear polynomial. Let…

A positive integer k is a length of a polynomial if that polynomial factors into a product of k irreducible polynomials. We find the set of lengths of polynomials of the form x^n in R[x], where (R, m) is an Artinian local ring with m^2 = 0.

We prove the following statement. Let $f\in\mathbb{R}[x_1,\ldots,x_d]$, for some $d\ge 3$, and assume that $f$ depends non-trivially in each of $x_1,\ldots,x_d$. Then one of the following holds. (i) For every finite sets…

We show that if two monic polynomials with integer coefficients have square-free resultant, then all positive divisors of the resultant arise as the greatest common divisor of the values of the two polynomials at a suitable integer.

We prove that for every positive integer $k$ there exist an inclusion-exclusion polynomial $Q_{\{q_1,q_2,...,q_k\}}$ with the height at least $c^{2^k}\prod_{j=1}^{k-2}q_j^{2^{k-j-1}-1}$, where $c$ is a positive constant and…

We study multiplicative dependence between terms of the $k$-generalized Pell sequence $(P_n^{(k)})_{n\ge 2-k}$, defined by the linear recurrence \[ P_n^{(k)} = 2P_{n-1}^{(k)} + P_{n-2}^{(k)} + \dots + P_{n-k}^{(k)}, \] with initial…

We contribute to the exceptional APN conjecture by showing that no polynomial of degree m = 2 r (2 {\ell} + 1) where gcd(r, {\ell}) 2, r 2, {\ell} 1 with a nonzero second leading coefficient can be APN over infinitely many extensions of the…

We present an elementary proof that the Schur polynomial corresponding to an increasing sequence of exponents (c_0,..., c_{n-1}) with c_0 = 0 is irreducible over every field of characteristic p whenever the numbers d_i = c_{i+1} - c_i are…

Let $(L_n^{(k)})_{n\geq 2-k}$ be the sequence of $k$-generalized Lucas numbers for some fixed integer $k\ge 2$, whose first $k$ terms are $0,\;\ldots\;,\;0,\;2,\;1$ and each term afterward is the sum of the preceding $k$ terms. In this…

Let $K/\mathbb Q$ be a finite Galois extension. Let $\chi_1,\ldots,\chi_r$ be $r\geq 1$ distinct characters of the Galois group with the associated Artin L-functions $L(s,\chi_1),\ldots, L(s,\chi_r)$. Let $m\geq 0$. We prove that the…

An independent set in a graph is a set of pairwise non-adjacent vertices. The independence number $\alpha{(G)}$ is the size of a maximum independent set in the graph $G$. The independence polynomial of a graph is the generating function for…

We discuss the existence of zero coefficients in the powers of the determinant polynomial of order $n$. D. G. Glynn proved that the coefficients of the $m$th power of the determinant polynomial are all nonzero, if $m = p-1$ with a prime…

Let $E$ be an elliptic curve, with identity $O$, and let $C$ be a cyclic subgroup of odd order $N$, over an algebraically closed field $k$ with $\operatorname{char} k \nmid N$. For $P \in C$, let $s_P$ be a rational function with divisor $N…

We prove that all polynomials in several variables can be decomposed as the sums of $k$th powers: $P(x_1,...,x_n) = Q_1(x_1,...,x_n)^k+...+ Q_s(x_1,...,x_n)^k$, provided that elements of the base field are themselves sums of $k$th powers.…