Related papers: On the maximum rank of a real binary form

Let $Q_1,...,Q_r\in \mathbb{Z}[x]$ be polynomials having $0$ as a root. Let $f(x,y)\in\mathbb{Z}[x,y]$ be a homogeneous polynomial with factorization $f(x,y)=f_1(x,y)^{e_1}\cdots f_u(x,y)^{e_u}$, where $f_i(x,y)$ are irreducible homogeneous…

Univariate polynomial root-finding is a classical subject, still important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the…

We provide upper bounds on the total number of irreducible factors, and in particular irreducibility criteria for some classes of bivariate polynomials $f(x,y)$ over an arbitrary field $\mathbb{K}$. Our results rely on information on the…

Suppose $I$ is an ideal of a polynomial ring over a field, $I\subseteq k[x_1,\ldots,x_n]$, and whenever $fg\in I$ with degree $\leq b$, then either $f\in I$ or $g\in I$. When $b$ is sufficiently large, it follows that $I$ is prime.…

We discuss several conjectures about the real-rootedness of polynomials whose coefficients are determinants of coefficients of a real-rooted polynomial. We also consider some questions about matrices generalizing totally positive matrices,…

We prove that the closure of the real roots of all-terminal reliability polynomials is exactly $[-1,0] \cup \{1\}$, resolving a conjecture of Brown and McMullin and refining the corresponding density result for multigraphs due to Brown and…

We give a complete conjectural formula for the number $e_r(d,m)$ of maximum possible ${\mathbb{F}}q$-rational points on a projective algebraic variety defined by $r$ linearly independent homogeneous polynomial equations of degree $d$ in…

Let $\mathbb{F}_q$ be a finite field with $q$ elements, $f \in \mathbb{F}_q[x_1, \dots, x_n]$ a polynomial in $n$ variables and let us denote by $N(f)$ the number of roots of $f$ in $\mathbb{F}_q^n$. %Many authors, such as Wei Cao and Kung…

We prove that in each degree divisible by 2 or 3, there are infinitely many totally real number fields that require universal quadratic forms to have arbitrarily large rank.

We first show the existence of an effective determinantal representation for any univariate polynomial with real coefficients. Then, we more precisely establish that any univariate polynomial with real coefficients has an effective…

Erd\"os and Obl\'ath proved that the equation $n!\pm m!=x^p$ has only finitely many integer solutions. More general, under the ABC-conjecture, Luca showed that $P(x)=An!+Bm!$ has finitely many integer solutions for polynomials of degree…

We state a kind of Euclidian division theorem: given a polynomial P(x) and a divisor d of the degree of P, there exist polynomials h(x),Q(x),R(x) such that P(x) = h(Q(x)) +R(x), with deg h=d. Under some conditions h,Q,R are unique, and Q is…

A real symmetric matrix $M$ is completely positive semidefinite if it admits a Gram representation by (Hermitian) positive semidefinite matrices of any size $d$. The smallest such $d$ is called the (complex) completely positive semidefinite…

Let $c(x_1,...,x_d)$ be a multihomogeneous central polynomial for the $n\times n$ matrix algebra $M_n(K)$ over an infinite field $K$ of positive characteristic $p$. We show that there exists a multihomogeneous polynomial $c_0(x_1,...,x_d)$…

In this paper we investigate the following related problems: (A) the separation of $p$-adic roots of integer polynomials of a fixed degree and bounded height; and (B) counting integer polynomials of a fixed degree and bounded height with…

Given any polynomial with real coefficients, the existence of a real quadratic polynomial factor is proven using only basic real analysis. The aim is to provide an approachable proof to anybody who is familiar with the least upper bound…

Given a nonnegative matrix M with rational entries, we consider two quantities: the usual positive semidefinite (psd) rank, where the matrix is factored through the cone of real symmetric psd matrices, and the rational-restricted psd rank,…

We prove that the (hermitian) rank of $QP^d$ is bounded from below by the rank of $P^d$ whenever $Q$ is not identically zero and real-analytic in a neighborhood of some point on the zero set of $P$ in $\mathbb{C}^n$ and $P$ is a polynomial…

Below we discuss the partition of the space of real univariate polynomials according to the number of positive and negative roots and signs of the coefficients. We present several series of non-realizable combinations of signs together with…

For a real degree $d$ polynomial $P$ with all nonvanishing coefficients, with $c$ sign changes and $p$ sign preservations in the sequence of its coefficients ($c+p=d$), Descartes' rule of signs says that $P$ has $pos\leq c$ positive and…