Quantifier Elimination over Finite Fields Using Gr\"obner Bases

We give an algebraic quantifier elimination algorithm for the first-order theory over any given finite field using Gr\"obner basis methods. The algorithm relies on the strong Nullstellensatz and properties of elimination ideals over finite fields. We analyze the theoretical complexity of the algorithm and show its application in the formal analysis of a biological controller model.Comment: A shorter version is to appear in International Conference on Algebraic Informatics 201

Alon's Nullstellensatz for multisets

Alon's combinatorial Nullstellensatz (Theorem 1.1 from \cite{Alon1}) is one of the most powerful algebraic tools in combinatorics, with a diverse array of applications. Let \F be a field, $S_1,S_2,..., S_n$ be finite nonempty subsets of \F. Alon's theorem is a specialized, precise version of the Hilbertsche Nullstellensatz for the ideal of all polynomial functions vanishing on the set S=S_1\times S_2\times ... \times S_n\subseteq \F^n. From this Alon deduces a simple and amazingly widely applicable nonvanishing criterion (Theorem 1.2 in \cite{Alon1}). It provides a sufficient condition for a polynomial $f(x_1,...,x_n)$ which guarantees that $f$ is not identically zero on the set $S$. In this paper we extend these two results from sets of points to multisets. We give two different proofs of the generalized nonvanishing theorem. We extend some of the known applications of the original nonvanishing theorem to a setting allowing multiplicities, including the theorem of Alon and F\"uredi on the hyperplane coverings of discrete cubes.Comment: Submitted to the journal Combinatorica of the J\'anos Bolyai Mathematical Society on August 5, 201

Computation with Polynomial Equations and Inequalities arising in Combinatorial Optimization

The purpose of this note is to survey a methodology to solve systems of polynomial equations and inequalities. The techniques we discuss use the algebra of multivariate polynomials with coefficients over a field to create large-scale linear algebra or semidefinite programming relaxations of many kinds of feasibility or optimization questions. We are particularly interested in problems arising in combinatorial optimization.Comment: 28 pages, survey pape