Related papers: Generic Bernstein-Sato polynomial on an irreducibl…

We give a constructive proof of the general Nullstellensatz: a univariate polynomial ring over a commutative Jacobson ring is Jacobson. This theorem implies that every finitely generated algebra over a zero-dimensional ring or the ring of…

The notion of symmetry in polynomial rings with several indeterminates is generalized to polynomial rings over finite fields. Families of extensions of the projective line over a finite field of constants possessing this property are…

A key property of an algebraic variety is whether it is absolutely irreducible, meaning that it remains irreducible over the algebraic closure of its defining field, and determining absolute irreducibility is important in algebraic geometry…

For sets $A, B\subset \mathbb N$, their sumset is $A + B := \{a+b: a\in A, b\in B\}$. If we cannot write a set $C$ as $C = A+B$ with $|A|, |B|\geq 2$, then we say that $C$ is $\textit{irreducible}$. The question of whether a given set $C$…

We study the problem of representing multivariate polynomials with rational coefficients, which are nonnegative and strictly positive on finite semialgebraic sets, using rational sums of squares. We focus on the case of finite semialgebraic…

Let $f_1$ and $f_2$ be two semi-universal deformations of quasi homogeneous polynomials in two variables respectively for the weight vectors $\rho_1$ and $\rho_2$ such that they satisfy similar conditions to that of semi quasi homogeneous…

Let K be an algebraically closed field of characteristic zero and let f(x,y) be a nonzero polynomial of K[x,y]. We prove that if the generic element of the family $(f-\lambda)\_{\lambda}$ is a rational polynomial, and if the Jacobian J(f,g)…

Given a separable nonconstant polynomial $f(x)$ with integer coefficients, we consider the set $S$ consisting of the squarefree parts of all the rational values of $f(x)$, and study its behavior modulo primes. Fixing a prime $p$, we…

Let $\mathbb{F}_q$ denote the finite field of characteristic $p$ and order $q$. Let $\mathbb{Z}_q$ denote the unramified extension of the $p$-adic rational integers $\mathbb{Z}_p$ with residue field $\mathbb{F}_q$. Given two positive…

Theorem. An irreducible cubic polynomial with rational coefficients has a root in a one step radical extension of Q if and only if the discriminate is a square of a rational number. Theorem. An irreducible polynomial x^4+px^2+qx+s with…

The paper studies the generic complex 1-dimensional polynomial vector fields of the form $iP(z)\frac{\partial}{\partial z}$, where $P$ is a polynomial with real coefficients, under topological orbital equivalence preserving the separatrices…

We consider the problem of defining polynomials over function fields of positive characteristic. Among other results, we show that the following assertions are true. 1. Let $\G_p$ be an algebraic extension of a field of $p$ elements and…

We generalize Block-G\"ottsche polynomials, originally defined for toric del Pezzo surfaces, to arbitrary surfaces. To do this, we show that these polynomials arise as special cases of BPS polynomials, defined for any surface $S$ as Laurent…

This paper deals with properties of the algebraic variety defined as the set of zeros of a "typical" sequence of polynomials. We consider various types of "nice" varieties: set-theoretic and ideal-theoretic complete intersections,…

Let $\mathbb{F}_q$ denote the finite field of $q$ elements and $\mathbb{F}_{q^n}$ the degree $n$ extension of $\mathbb{F}_q$. A normal basis of $\mathbb{F}_{q^n}$ over $\mathbb{F} _q$ is a basis of the form…

Regarding polynomial functions on a subset $S$ of a non-commutative ring $R$, that is, functions induced by polynomials in $R[x]$ (whose variable commutes with the coefficients), we show connections between, on one hand, sets $S$ such that…

The Schinzel hypothesis essentially claims that finitely many irreducible polynomials in one variable over Z simultaneously assume infinitely many prime values unless there is an obvious reason why this is impossible. We prove that under a…

We present a variant of the Peskine--Szpiro Acyclicity Lemma, and hence a way to certify exactness of a complex of finite modules over a large class of (possibly) noncommutative rings. Specifically, over the class of Auslander regular…

We give a combinatorial description of the roots of the Bernstein-Sato polynomial of a monomial ideal using the Newton polyhedron and some semigroups associated to the ideal.

In 2006, Budur, Musta\c{t}\v{a} and Saito introduced the notion of Bernstein-Sato polynomial of an arbitrary scheme of finite type over fields of characteristic zero. Because of the strong monodromy conjecture, it should have a…