Related papers: Generalised polynomials and integer powers

A natural number is a binary $k$'th power if its binary representation consists of $k$ consecutive identical blocks. We prove an analogue of Waring's theorem for sums of binary $k$'th powers. More precisely, we show that for each integer $k…

In this article, we consider polynomials of the form $f(x)=a_0+a_{n_1}x^{n_1}+a_{n_2}x^{n_2}+\dots+a_{n_r}x^{n_r}\in \mathbb{Z}[x],$ where $|a_0|\ge |a_{n_1}|+\dots+|a_{n_r}|,$ $|a_0|$ is a prime power and $|a_0|\nmid |a_{n_1}a_{n_r}|$. We…

Let $f(x)\in \mathbb{Z}[x]$ be an irreducible polynomial of degree $d\ge 1$. Let $k\ge2$ be an integer. The number of integers $n$ such that $f(n)$ is $k$-free is widely studied in the literature. In principle, one expects that $f(n)$ is…

For a polynomial in several variables depending on some parameters, we discuss some results to the effect that for almost all values of the parameters the polynomial is irreducible. In particular we recast in this perspective some results…

We produce a new family of polynomials f(x) over fields K of characteristic 2 which are exceptional, in the sense that f(x)-f(y) has no absolutely irreducible factors in K[x,y] besides the scalar multiples of x-y; when K is finite, this…

We study the algebraic properties of Generalized Laguerre Polynomials for negative integral values of the parameter. For integers $r,n\geq 0$, we conjecture that $L_n^{(-1-n-r)}(x) = \sum_{j=0}^n \binom{n-j+r}{n-j}x^j/j!$ is a…

Let K be a number field and let A be its ring of integers. Let G be a connected, noncommutative, absolutely almost simple algebraic K-group. If the K-rank of G equals 2, then G(A[t]) is not finitely presented.

Let $G$ be a topological Abelian semigroup with unit, let $E$ be a Banach space, and let $C(G,E)$ denote the set of continuous functions $f\colon G\to E$. A function $f\in C(G,E)$ is a generalized polynomial, if there is an $n\ge 0$ such…

We consider polynomials $Q:=\sum _{j=0}^da_jx^j$, $a_j\in \mathbb{R}^*$, with all roots real. When the {\em sign pattern} $\sigma (Q):=({\rm sgn}(a_d),{\rm sgn}(a_{d-1})$, $\ldots$, ${\rm sgn}(a_0))$ has $\tilde{c}$ sign changes, the…

Given a natural number $n \geq 2$, an integer $k$ and for a judiciously chosen $l = l(n)$ we give necessary and sufficient conditions for the polynomial $f_{n,k} = \big( \sum_{i=1}^{l} x_{i}^{n} \big) - k$ to have roots modulo every…

A new polynomial sieve is presented and used to show that almost all integers have at most one representation as a sum of two values of a given polynomial of degree at least 3.

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.

In this paper we consider generalized monomial functions $f, g\colon \mathbb{F}\to \mathbb{C}$ (of possibly different degree) that also fulfill \[ f(P(x))= Q(g(x)) \qquad \left(x\in \mathbb{F}\right), \] where $P\in \mathbb{F}[x]$ and $Q\in…

Suppose that $\langle f_n \rangle$ is a sequence of polynomials, $\langle f_n^{(k)}(0)\rangle$ converges for every non-negative integer $k$, and that the limit is not $0$ for some $k$. It is shown that if all the zeros of $f_1, f_2, \dots$…

In this paper we prove a characterization of continuity for polynomials on a normed space. Namely, we prove that a polynomial is continuous if and only if it maps compact sets into compact sets. We also provide a partial answer to the…

Denote by $\Gamma$ the set of pointwise good sequences. Those are sequences of real numbers $(a_k)$ such that for any measure preserving flow $(U_t)_{t\in \mathbb R}$ on a probability space and for any $f\in L^\infty$, the averages…

For a fixed integer $k \ge 0$, consider representations of positive integers as sums of binomial coefficients of the form $\binom{n}{k}$. While exact minimal bounds for the number of required summands are known only in a few low-dimensional…

Answering a question asked by Hsia and Tucker in their paper on the finiteness of greatest common divisors of iterates of polynomials, we prove that if $f, g \in \mathbb{C}(X)$ are compositionally independent rational functions and $c \in…

We study an infinite class of sequences of sparse polynomials that have binomial coefficients both as exponents and as coefficients. This generalizes a sequence of sparse polynomials which arises in a natural way as graph theoretic…

We study the polynomial algebra (over a ring containing the rationals) in an n by m matrix of variables, and subject to the relation that says that the product of any two variables in the same column is zero. We show that the sub-algebra of…