The purpose of the article is to explain what these theorems mean, and also to give an outline of the proof of the first one. The intended audience is mathematicians with an interest in finite fields, but no especial expertise on the vast literature which surrounds the topic of equations over finite fields. By way of motivation, we will begin by giving an indication of the historical significan...

Introduction 1. Comparison of Density Conditions 2. Curves 3. Conic Bundles over P 4. Smooth Cubic Hypersurfaces 5. Smooth Complete Intersections of Two Quadrics in P 6. Elliptic Surfaces 7. Some Abelian Varieties 8. Some Kummer Surfaces 9. Some Other K3 Surface Examples References The aim of this article is to provoke a discussion concerning the general nature of the topological closure of the...

In this paper we study a general near-Hamiltonian polynomial system on the plane. We suppose the unperturbed system has a family of periodic orbits surrounding a center point and obtain some sufficient conditions to find the cyclicity of the perturbed system at the center or a periodic orbit. In particular, we prove that for almost all polynomial Hamiltonian systems the perturbed systems with p...

In this paper, a multiplicity preserving triangular set decomposition algorithm is proposed for a system of two polynomials. The algorithm decomposes the variety defined by the polynomial system into unmixed components represented by triangular sets, which may have negative multiplicities. In the bivariate case, we give a complete algorithm to decompose the system into multiplicity preserving t...

We consider the solution of spectral problems with elliptic potentials in the framework of the Hermite ansatz. We show that the search for potentials and their spectral characteristics is reduced to a system of polynomial equations solvable by the Gröbner bases method and others. New potentials and corresponding solutions of the Sawada–Kotera, Kaup–Kupershmidt, Boussinesq equations are found.

This paper provides a method to describe and solve a combinatorics problem using systems of polynomial equations. These systems, however, are too large to be solved by hand. The goal of this paper is to give the reader two techniques to solve these systems. The first technique uses Buchberger’s Algorithm to find a Gröbner basis for the system. The second technique addresses and solves the probl...

Solving a system of polynomial equations is a ubiquitous problem in the applications of mathematics. Until recently, it has been hopeless to find explicit solutions to such systems, and mathematics has instead developed deep and powerful theories about the solutions to polynomial equations. Enumerative Geometry is concerned with counting the number of solutions when the polynomials come from a ...

Given a polynomial equation of degree d over the complex domain, the Fundamental Theorem of Algebra tells us that there are d solutions, assuming that the solutions are counted by multiplicity. These solutions can be approximated by deforming a standard «th degree equation into the given equation, and following the solutions through the deformation. This is called the homotopy method. The Funda...

We address the problem of polynomial time solving univariate modular equations with mutually co-prime moduli. For a given system of equations we determine up to which size the common roots can be calculated efficiently. We further determine the minimum number of equations which suffice for a recovery of all common roots. The result that we obtain is superior to Håstad’s original RSA broadcast a...

A new technique for the Geometry of Numbers is exhibited. This technique provides sharp estimates on the number of bounded height rational rational points in subsets of projective space whose “projective cone” is semi–algebraic. This technique improves existing techniques as the one introduced by H. Davenport in [15]. As main outcome, we conclude that systems of rational polynomial equations of...