Related papers: On a Frobenius problem for polynomials

Primitive polynomials over finite fields are crucial for various domains of computer science, including classical pseudo-random number generation, coding theory and post-quantum cryptography. Nevertheless, the pursuit of an efficient…

In this extended abstract we deal with the relations between the numerical/diophantine approximation and the symbolic/algebraic geometry approachs to solving of multivariate diophentine polynomial systems, obtaining several consecuences…

Let $K$ be a number field and $r$ an integer. Given an elliptic curve $E$, defined over $K$, we consider the problem of counting the number of degree two prime ideals of $K$ with trace of Frobenius equal to $r$. Under certain restrictions…

We reduce the Nowicki conjecture on the Weitzenb\"ock derivation of polynomial algebras to well-known problem of the classical invariant theory.

Permutation polynomials with few terms attracts researchers' interest in recent years due to their simple algebraic form and some additional extraordinary properties. In this paper, by analyzing the quadratic factors of a fifth-degree…

We compute the Groebner basis of a system of polynomial equations related to the Jacobian conjecture, and describe completely the solution set.

We study how certain invariants of numerical semigroups relate to the number of second kind gaps. Furthermore, given two fixed non-negative integers F and k, we provide an algorithm to compute all the numerical semigroups whose Frobenius…

We show that it makes sense to speak of THE Frobenius manifold attached to a convenient and nondegenerate Laurent polynomial

Let $K$ be an algebraically closed field with an absolute value. This note gives an elementary proof of the classical result that the roots of a polynomial with coefficients in $K$ are continuous functions of the coefficients of the…

We give an effective solution of the conjugacy problem for two by two matrices over the polynomial ring in one variable over a finite field.

We discuss the explicit construction of the Schroedinger equations admitting a representation through some family of general polynomials. Almost all solvable quantum potentials are shown to be generated by this approach. Some generalization…

Let $f(x_1,...,x_k)$ be a polynomial over a field $K$. This paper considers such questions as the enumeration of the number of nonzero coefficients of $f$ or of the number of coefficients equal to $\alpha\in K^*$. For instance, if $K=\ff_q$…

The following result, a consequence of Dumas criterion for irreducibility of polynomials over integers, is generally proved using the notion of Newton diagram: Let $f(x)$ be a polynomial with integer coefficients and $k$ be a positive…

A central question in Arithmetic geometry is to determine for which polynomials $f \in \mathbb{Z}[t]$ and which number fields $K$ the Hasse principle holds for the affine equation $f(t) = N_{K/\mathbb{Q}}(\boldsymbol{x}) \neq 0$. Whilst…

We present a method for the solution of polynomial equations. We do not intend to present one more method among several others, because today there are many excellent methods. Our main aim is educational. Here we attempt to present a method…

We prove that the Newton polygons of Frobenius on the crystalline cohomology of proper smooth varieties satisfy a symmetry that results, in the case of projective smooth varieties, from Poincar\'e duality and the hard Lefschetz theorem. As…

We use ideas from our previous work to obtain some theorems that will allow us to obtain the integer solution of a quadratic polynomial in two variables that represents a natural number

We explore two questions about pseudo-polynomials, which are functions $f:\mathbb N \to \mathbb Z$ such that $k$ divides $f(n+k) - f(n)$ for all $n,k$. First, for certain arbitrarily sparse sets $R$, we construct pseudo-polynomials $f$ with…

We consider the average-case complexity of some otherwise undecidable or open Diophantine problems. More precisely, we show that the following two problems can be solved in the complexity class PSPACE: (I) Given polynomials f_1,...,f_m in…

We prove a completeness result for a class of polynomial solutions of the wave equation called wave polynomials and construct generalized wave polynomials, solutions of the Klein-Gordon equation with a variable coefficient. Using the…