相关论文: The number of real zeros of polynomials with const…

We prove the complete monotonicity on $(0,\infty)^n$ for suitable inverse powers of the spanning-tree polynomials of graphs and, more generally, of the basis generating polynomials of certain classes of matroids. This generalizes a result…

I show that there exist universal constants $C(r) < \infty$ such that, for all loopless graphs $G$ of maximum degree $\le r$, the zeros (real or complex) of the chromatic polynomial $P_G(q)$ lie in the disc $|q| < C(r)$. Furthermore, $C(r)…

Let $p(z)=a_0+a_1z+a_2z^2+a_3z^3+\cdots+a_nz^n$ be a polynomial of degree $n$ having no zeros in the unit disk. ~Then it is well known that for $R\geq 1,$ $\displaystyle{\max_{|z|=R}|p(z)|}\leq…

We prove the classical result, which goes back at least to Fourier, that a polynomial with real coefficients has all zeros real and distinct if and only if the polynomial and also all of its nonconstant derivatives have only negative minima…

We consider polynomials of the form t^n-1 and determine when members of this family have a divisor of every degree in Z[t]. With F(x) defined to be the number of such integers up to x, we prove the existence of two positive constants c_1…

This paper investigates the zero distribution of a sequence of polynomials $\left\{ P_{m}(z)\right\} _{m=0}^{\infty}$ generated by the reciprocal of $1+ct+B(z)t^{2}+A(z)t^{3}$ where $c\in\mathbb{R}$ and $A(z)$, $B(z)$ are real linear…

This paper discusses the location of zeros of polynomials in a polynomial sequence $\{P_n(z)\}$ generated by a three-term recurrence relation of the form $P_n(z)+ B(z)P_{n-1}(z) +A(z) P_{n-k}(z)=0$ with $k>2$ and the standard initial…

Given $H\subseteq \mathbb{C}$ two natural objects to study are the set of zeros of polynomials with coefficients in $H$, $$\{z\in \mathbb{C}: \exists k>0,\, \exists (a_n)\in H^{k+1}, \sum_{n=0}^{k}a_{n}z^n=0\},$$ and the set of zeros of…

For each fixed value of $\beta$ in the range $-2<\beta<-1$ and $0<c<1$, we investigate interlacing properties of the zeros of polynomials of consecutive degree for $M_{n}(x;\beta,c)$ and $M_k(x,\beta+t,c)$, $k\in\{n-1,n,n+1\}$ and…

In 1931 Bohnenblust and Hille proved that for each m-homogeneous polynomial $\sum_{|\alpha| = m} a_\alpha z^\alpha$ on $\C^n$ the $\ell^{\frac{2m}{m+1}}$-norm of its coefficients is bounded from above by a constant $C_m$ (depending only on…

We show that there is an absolute constant $c > 1/2$ such that the Mahler measure of the Fekete polynomials $f_p$ of the form $$f_p(z) := \sum_{k=1}^{p-1}{\left( \frac kp \right)z^k}\,,$$ (where the coefficients are the usual Legendre…

Let $(A,\mathfrak{m})$ be a complete intersection of dimension $d \geq 1$ and codimension $c \geq 1$. Let $I$ be an $\mathfrak{m}$-primary ideal and let $M$ be a finitely generated $A$-module. For $i \geq 1$ let $\psi_i^I(M)$ be the degree…

We show that under the assumption of a 24-term version of Fermat's Last Theorem, there exists an absolute constant c > 0 such that if S is a set of n > n_0 positive integers satisfying |S.S| < n^(1+c), then the sumset S.S satisfies |S+S| >>…

Given a sequence of orthogonal polynomials $(p_n)_n$ with respect to a positive measure in the real line, we study the real zeros of finite combinations of $K+1$ consecutive orthogonal polynomials of the form $$…

For $0<\alpha<1$, we study the zeros of the sequence of polynomials $\left\{ P_{m}(z)\right\} _{m=0}^{\infty}$ generated by the reciprocal of $(1-t)^{\alpha}(1-2zt+t^{2})$, expanded as a power series in $t$. Equivalently, this sequence is…

We prove that there exists an absolute constant $c>0$ such that if an arithmetic progression $\cP$ modulo a prime number $p$ does not contain zero and has the cardinality less than $cp$, then it can not be represented as a product of two…

Let $f$ be an ordinary polynomial in $\mathbb{C}[z_1,..., z_n]$ with no negative exponents and with no factor of the form $z_1^{\alpha_1}... z_n^{\alpha_n}$ where $\alpha_i$ are non zero natural integer. If we assume in addicting that $f$…

For each $\alpha>0$ and $A(z),B(z)\in\mathbb{C}[z]$, we study the zero distribution of the sequence of polynomials $\left\{ P_{m}^{(\alpha)}(z)\right\} _{m=0}^{\infty}$ generated by $(1+B(z)t+A(z)t^{3})^{-\alpha}$. We show that for large…

Given coprime integers $k, \ell$ with $k > \ell \geqslant 1$ and arbitrary complex polynomials $A(z), B(z)$ with $\deg(A(z)B(z))\geqslant 1$, we consider the polynomial sequence $\{P_n(z)\}$ satisfying a three-term recurrence…

We show that there exists an absolute constant $c > 0$, such that, for any finite set $A$ of quaternions, \[ \max\{|A+A, |AA| \} \gtrsim |A|^{4/3 + c}. \] This generalizes a sum-product bound for real numbers proved by Konyagin and…