Related papers: Integral polynomials with small discriminants and …

We study lower bounds for the norm of the product of polynomials and their applications to the so called \emph{plank problem.} We are particularly interested in polynomials on finite dimensional Banach spaces, in which case our results…

Low-degree polynomials have emerged as a powerful paradigm for providing evidence of statistical-computational gaps across a variety of high-dimensional statistical models [Wein25]. For detection problems -- where the goal is to test a…

The problem of finding the distance from a given $n \times n$ matrix polynomial of degree $k$ to the set of matrix polynomials having the elementary divisor $(\lambda-\lambda_0)^j, \, j \geqslant r,$ for a fixed scalar $\lambda_0$ and $2…

Let $D$ be a subset of a finite commutative ring $R$ with identity. Let $f(x)\in R[x]$ be a polynomial of positive degree $d$. For integer $0\leq k \leq |D|$, we study the number $N_f(D,k,b)$ of $k$-subsets $S\subseteq D$ such that…

Define B(n) to be the largest height of a polynomial in $\mathbb{Z}[x]$ dividing $x^n-1$. We formulate a number of conjectures related to the value of B(n) when $n$ is of a prescribed form. Additionally, we prove a lower bound for…

Differential (Ore) type polynomials with "approximate" polynomial coefficients are introduced. These provide an effective notion of approximate differential operators, with a strong algebraic structure. We introduce the approximate Greatest…

We consider orthogonal polynomials with respect to the weight $|z^2+a^2|^{cN}e^{-N|z|^2}$ in the whole complex plane. We obtain strong asymptotics and the limiting normalized zero counting measure (mother body) of the orthogonal polynomials…

The paper considers the problem of finding the largest possible set P(n), a subset of the set N of the natural numbers, with the property that a number is in P(n) if and only if it is a sum of n distinct naturals all in P(n) or none in…

The triangle of sorted binomial coefficients $\left\langle {n \atop k} \right\rangle = \binom{n}{\lfloor \frac{n - k}{2} \rfloor}$ for $0 \leq k \leq n$ has appeared several times in recent combinatorial works but has evaded dedicated…

We establish asymptotic upper bounds on the number of zeros modulo $p$ of certain polynomials with integer coefficients, with $p$ prime numbers arbitrarily large. The polynomials we consider have degree of size $p$ and are obtained by…

In this work, we consider rational ordinary differential equations dy/dx = Q(x,y)/P(x,y), with Q(x,y) and P(x,y) coprime polynomials with real coefficients. We give a method to construct equations of this type for which a first integral can…

Derivative polynomials in two variables are defined by repeated differentiation of the tangent and secant functions. We establish the connections between the coefficients of these derivative polynomials and the numbers of interior and left…

We adapt the arguments of Marcus, Spielman and Srivastava in their proof of the Kadison-Singer problem to prove improved paving estimates. Working with Anderson's paving formulation of Kadison-Singer instead of Weaver's vector balancing…

Differential resultant formulas are defined, for a system $\mathcal{P}$ of $n$ ordinary Laurent differential polynomials in $n-1$ differential variables. These are determinants of coefficient matrices of an extended system of polynomials…

This paper is devoted to the explicit construction of optimal designs for discrimination between two polynomial regression models of degree $n-2$ and $n$. In a fundamental paper, Atkinson and Fedorov [Biometrika 62 (1975a) 57--70] proposed…

This paper focuses on the study of a mathematical program with equilibrium constraints, where the objective and the constraint functions are all polynomials. We present a method for finding its global minimizers and global minimum using a…

In this paper we derive aggregate separation bounds, named after Davenport-Mahler-Mignotte (\dmm), on the isolated roots of polynomial systems, specifically on the minimum distance between any two such roots. The bounds exploit the…

In this paper we study pairs of polynomials with a given factorization pattern and such that the degree of their difference attains its minimum. We call such pairs of polynomials Davenport--Zannier pairs, or DZ-pairs for short. The paper is…

For the family of polynomials in one variable $P:=x^n+a_1x^{n-1}+\cdots +a_n$, $n\geq 4$, we consider its higher-order discriminant sets $\{ \tilde{D}_m=0\}$, where $\tilde{D}_m:=$Res$(P,P^{(m)})$, $m=2$, $\ldots$, $n-2$, and their…

We propose a method for constructing systems of polynomial equations that define submanifolds of degenerate binary forms of an arbitrary degeneracy degree. It is appropriate to call these systems of equations "higher discriminants".