Related papers: Paucity phenomena for polynomial products

Let $\mathbb{P}= \{P_1, \cdots, P_{k}\in \mathbb{R}[y]\}$ be a collection of polynomials with distinct degrees and zero constant terms. We proved that there exists $\epsilon=\epsilon(\mathbb{P})>0$ such that, for any compact set $E \subset…

Fix an odd prime $p$. If $r$ is a positive integer and $f$ a polynomial with coefficients in $\mathbb{F}_{p^r}$, let $P_{p,r}(f)$ be the proportion of $\mathbb{P}^1(\mathbb{F}_{p^r})$ that is periodic with respect to $f$. We show that as…

Using a recent result of Salberger, we establish the paucity of non-trivial positive integer solutions to a certain system of diagonal Diophantine equations.

We prove that if $P(X) \in \mathbb{Z}[X]$ is an integer polynomial of degree $n$ and having $P(0) = 1$, then either $P(X)$ is a product of cyclotomic polynomials, or else at least one of the complex roots of $P$ belongs to the disk $|z|…

There is a natural pluripotential-theoretic extremal function V_{K,Q} associated to a closed subset K of C^m and a real-valued, continuous function Q on K. We define random polynomials H_n whose coefficients with respect to a related…

We propose a framework to study models of computation of indeterministic data, represented by abstract "distributions". In these distributions, probabilities are replaced by "amplitudes" drawn from a fixed semi-ring $S$, of which the…

In this article, we consider the following family of random trigonometric polynomials $p_n(t,Y)=\sum_{k=1}^n Y_{k,1} \cos(kt)+Y_{k,2}\sin(kt)$ for a given sequence of i.i.d. random variables $\{Y_{k,1},Y_{k,2}\}_{k\ge 1}$ which are centered…

This paper studies Positivstellens\"atze and moment problems for sets $K$ that are given by universal quantifiers. Let $Q$ be a closed set and let $g = (g_1,...,g_s)$ be a tuple of polynomials in two vector variables $x$ and $y$. Then $K$…

Wolstenholme's type summations involve certain powers of all residues $k$ modulo some prime number $p$. We first consider the sums of double or triple products of certain powers of all residues, e.g., the sums of the terms $(a+k)^m(b+k)^n$…

We prove a solvability theorem for the Stieltjes moment problem on $R^d$ which is based on the multivariate Stieltjes condition $\sum_{n=1}^\infty L(x_j^n)^{-1/(2n)}=+\infty$, $j=1,\dots,d.$ This result is applied to derive a new…

Let $\varphi_1,\ldots ,\varphi_r\in \mathbb Z[z_1,\ldots z_k]$ be integral linear combinations of elementary symmetric polynomials with $\text{deg}(\varphi_j)=k_j$ $(1\le j\le r)$, where $1\le k_1<k_2<\ldots <k_r=k$. Subject to the…

Given a polynomial $Q(x_1,\cdots, x_t)=\lambda_1 x_1^{k_1}+\cdots +\lambda_t x_t^{k_t}$, for every $c\in \mathbb{Z}$ and $n\geq 2$, we study the number of solutions $N_J(Q;c,n)$ of the congruence equation $Q(x_1,\cdots, x_t)\equiv…

The computational complexity of the partition, 0-1 subset sum, unbounded subset sum, 0-1 knapsack and unbounded knapsack problems and their multiple variants were studied in numerous papers in the past where all the weights and profits were…

Let $f(x) \in \mathbb{Z}[x]$. Set $f_{0}(x) = x$ and, for $n \geq 1$, define $f_{n}(x)$ $=$ $f(f_{n-1}(x))$. We describe several infinite families of polynomials for which the infinite product \prod_{n=0}^{\infty} (1 + \frac{1}{f_{n}(x)})…

This paper is concerned with the following fractional $p$-Kirchhoff equation \begin{eqnarray*} \varepsilon ^{sp}M\left( {\varepsilon ^{sp - N}}\iint_{\mathbb{R}^{2N}}\frac{{{{\left| {u(x) - u(y)} \right|}^p}}}{{{{\left| {x - y} \right|}^{N…

We consider a wide class of summatory functions F{f;N,p^m}=\sum_{k\leq N}f(p^m k), m\in \mathbb Z_+\cup {0}, associated with the multiplicative arithmetic functions f of a scaled variable k\in \mathbb Z_+, where p is a prime number.…

Let $\ast_P$ be a product on $l_{\rm{fin}}$ (a space of all finite sequences) associated with a fixed family $(P_n)_{n=0}^{\infty}$ of real polynomials on $\mathbb{R}$. In this article, using methods from the theory of generalized…

For products $P_N$ of $N$ random matrices of size $d \times d$, there is a natural notion of finite $N$ Lyapunov exponents $\{\mu_i\}_{i=1}^d$. In the case of standard Gaussian random matrices with real, complex or real quaternion elements,…

We show that for a random polynomial \[ F(X) = \sum_{n=1}^{N} f(n) X^{n-1}, \] where $f(n)$ is a random completely multiplicative function taking values in $\{\pm 1\}$, one has \[ \limsup_{N \to \infty} \mathbb{P}\big[F(X) \text{ is…

Consider a sequence of random polynomials $P_n(z) = \prod_{k=1}^{n}(z - X_k)$, where $\{X_k\}_k$ are i.i.d. random variables distributed uniformly on the unit disc $\mathbb{D}$. Let $\Lambda_n = \{z \in \mathbb{C}: |P_n(z)| < 1\}$ be the…