Related papers: The best possible quadratic refinement of Sendov's…

For a graph $G$ on $[n]$, the $k$-cut complex $\Delta_k(G)$ has facets $[n]\setminus T$, where $T$ ranges over the disconnected $k$-vertex induced subgraphs of $G$. Bayer, Denker, Jeli\'c Milutinovi\'c, Sundaram, and Xue proved that the…

We consider random polynomials whose coefficients are independent and identically distributed on the integers. We prove that if the coefficient distribution has bounded support and its probability to take any particular value is at most…

Let $Z\subset{\bf P}^{n-1}$ be a hypersurface such that the associated reduced hypersurface $Z_{\rm red}$ has only weighted homogeneous isolated singularities. In the case $Z$ is a reduced curve or $Z_{\rm red}$ has only homogeneous…

Assuming the four exponentials conjecture, Hansel and Safer showed that if a subset $S$ of the Gaussian integers is both $\alpha=-m+i $- and $\beta=-n+i$-recognizable, then it is syndetic, and they conjectured that $S$ must be eventually…

The Schwartz-Zippel Lemma states that if a low-degree multivariate polynomial with coefficients in a field is not zero everywhere in the field, then it has few roots on every finite subcube of the field. This fundamental fact about…

A number of authors have proven explicit versions of Lehmer's conjecture for polynomials whose coefficients are all congruent to 1 modulo m. We prove a similar result for polynomials f(X) that are divisible in (Z/mZ)[X] by a polynomial of…

In 1989, Rota made the following conjecture. Given $n$ bases $B_{1},\dots,B_{n}$ in an $n$-dimensional vector space $V$, one can always find $n$ disjoint bases of $V$, each containing exactly one element from each $B_{i}$ (we call such…

This is a study of the sequences of Betti numbers of finitely generated modules over a complete intersection local ring, $R$. The subsequences $\{\beta^R_{i}(M)\}_{i\geq 0}$ with even, respectively, odd $i$ are known to be eventually given…

The probability that a zero of a random real polynomial of increasing degree is real tends to zero. However, passing from polynomials to Laurent polynomials yields a surprising result: the probability that a root is real tends not to zero,…

We investigate structural properties of the cone of roots of relative Steiner polynomials of convex bodies. We prove that they are closed, monotonous with respect to the dimension, and that they cover the whole upper half-plane, except the…

We present a sufficient condition for a self-inversive polynomial to have a fixed number of roots on the complex unit circle. We also prove that these roots are simple when that condition is satisfied. This generalizes the condition found…

We define the second discriminant $D_2$ of a univariate polynomial $f$ of degree greater than $2$ as the product of the linear forms $2\,r_k-r_i-r_j$ for all triples of roots $r_i, r_k, r_j$ of $f$ with $i<j$ and $j\neq k, k\neq i$. $D_2$…

For any polynomial f with complex coefficients we find a remarkable subset of poles of the motivic zeta function. It is combinatorially determined by any log resolution and it admits an intrinsic interpretation in terms of contact loci of…

We prove that for every $\varepsilon>0$ and a nonnegative integer $\omega$ there exist primes $p_1,p_2,\ldots,p_\omega$ such that for $n=p_1p_2\ldots p_\omega$ the height of the cyclotomic polynomial $\Phi_n$ is at least…

Let $ P(z) $ be a polynomial of degree $ n $ having all zeros in $|z|\leq k$ where $k\leq 1,$ then it was proved by Dewan \textit{et al} that for every real or complex number $\alpha$ with $|\alpha|\geq k$ and each $r\geq 0$ $$…

Let $G$ be a connected graph, and let $\lambda_1$ and $\rho$ denote the spectral radius of $G$ and the universal cover of $G$, respectively. In \cite{Fri03}, Friedman has shown that almost every $n$-lift of $G$ has all of its new…

According to Casas Alvero conjecture, if a one variable polynomial of degree $n$ over a field of characteristic 0 is not prime with each of the $n-1$ first derivees, then it is of the form $c (X-r)^{n}$. Let $p$ be a prime number and an…

A tensor defined over a finite field $\mathbb{F}$ has low analytic rank if the distribution of its values differs significantly from the uniform distribution. An order $d$ tensor has partition rank 1 if it can be written as a product of two…

In this paper, we study polynomial norms, i.e. norms that are the $d^{\text{th}}$ root of a degree-$d$ homogeneous polynomial $f$. We first show that a necessary and sufficient condition for $f^{1/d}$ to be a norm is for $f$ to be strictly…

Given $n$ positive integers $a_1,a_2,\dots,a_n$, and a positive integer right hand side $\beta$, we consider the feasibility version of the subset sum problem which is the problem of determining whether a subset of $a_1,a_2,\dots,a_n$ adds…