Related papers: On the characterization of uniqueness polynomials:…

We investigate the computational complexity of deciding whether a given univariate integer polynomial p(x) has a factor q(x) satisfying specific additional constraints. When the only constraint imposed on q(x) is to have a degree smaller…

It is known that point searching in basic semialgebraic sets and the search for globally minimal points in polynomial optimization tasks can be carried out using $(s\,d)^{O(n)}$ arithmetic operations, where $n$ and $s$ are the numbers of…

We study the P versus NP problem through properties of functions and monoids, continuing the work of [3]. Here we consider inverse monoids whose properties and relationships determine whether P is different from NP, or whether injective…

The aim of this paper is a quantitative analysis of the solution set of a system of polynomial nonlinear differential equations, both in the ordinary and partial case. Therefore, we introduce the differential counting polynomial, a common…

We study an infinite class of sequences of sparse polynomials that have binomial coefficients both as exponents and as coefficients. This generalizes a sequence of sparse polynomials which arises in a natural way as graph theoretic…

We classify all post-critically finite unicritical polynomials defined over the maximal totally real algebraic extension of ${\mathbb Q}$. Two auxiliary results used in the proof of this result may be of some independent interest. The first…

A continuous selection of polynomial functions is a continuous function whose domain can be partitioned into finitely many pieces on which the function coincides with a polynomial. Given a set of finitely many polynomials, we show that…

Polynomial factorization is a fundamental problem in computational algebra. Over the past half century, a variety of algorithmic techniques have been developed to tackle different variants of this problem. In parallel, algebraic complexity…

The paper deals with a complex polynomial $H$ in two variables having - a generic highest homogeneous part (without multiple zero lines), - nonconstant lower terms. In particular, under these conditions the polynomial $H$ has at least two…

Solving polynomial equations is a subtask of polynomial optimization. This article introduces systems of such equations and the main approaches for solving them. We discuss critical point equations, algebraic varieties, and solution counts.…

We show that deciding whether a sparse univariate polynomial has a p-adic rational root can be done in NP for most inputs. We also prove a polynomial-time upper bound for trinomials with suitably generic p-adic Newton polygon. We thus…

We give a first-order definition of key polynomials, we show the links with previous definitions, that it is relevant to study key degrees, and to use a kind of valuations that we call partially multiplicative. We also prove or reprove…

We define dual-critical graphs as graphs having an acyclic orientation, where the indegrees are odd except for the unique source. We have very limited knowledge about the complexity of dual-criticality testing. By the definition the problem…

We obtain a new bound of certain double multiplicative character sums. We use this bound together with some other previously obtained results to obtain new algorithms for finding roots of polynomials modulo a prime $p$.

In pursuit of a deeper understanding of Boolean Promise Constraint Satisfaction Problems (PCSPs), we identify a class of problems with restricted structural complexity, which could serve as a promising candidate for complete…

A system of singular integral equations with monotone and concave nonlinearity in the subcritical case is investigated. The specified system and its scalar analog have direct applications in various areas of physics and biology. In…

In this article, we study critical points (zeros of derivative) of random polynomials. Take two deterministic sequences $\{a_n\}_{n\geq1}$ and $\{b_n\}_{n\geq1}$ of complex numbers whose limiting empirical measures are same. By choosing…

We survey three methods for proving that the characteristic polynomial of a finite lattice factors over the nonnegative integers and indicate how they have evolved recently. The first technique uses geometric ideas and is based on…

We give lower bounds for the degree of the discriminant with respect to y of separable polynomials f in K[x,y] over an algebraically closed field of characteristic zero. Depending on the invariants involved in the lower bound, we give a…

A recently-established necessary condition for polynomials that preserve the class of entrywise nonnegative matrices of a fixed order is shown to be necessary and sufficient for the class of nonnegative monomial matrices. Along the way, we…