Short Rational Functions for Toric Algebra and Applications

TL;DR

This paper introduces a novel method using short rational functions to efficiently encode and compute properties of toric ideals, enabling polynomial-time algorithms for Graver bases, Gröbner bases, and normal forms, with applications in algebra, integer programming, and statistics.

Contribution

It presents a new rational function encoding for toric ideals that allows polynomial-time computation of key algebraic structures and normal forms, improving efficiency over traditional methods.

Findings

Polynomial-time algorithms for Graver and Gröbner bases.

Efficient computation of normal forms.

Applications to Hilbert series, integer programming, and statistics.

Abstract

We encode the binomials belonging to the toric ideal $I_A$ associated with an integral $d \times n$ matrix $A$ using a short sum of rational functions as introduced by Barvinok \cite{bar,newbar}. Under the assumption that $d,n$ are fixed, this representation allows us to compute the Graver basis and the reduced Gr\"obner basis of the ideal $I_A$, with respect to any term order, in time polynomial in the size of the input. We also derive a polynomial time algorithm for normal form computation which replaces in this new encoding the usual reductions typical of the division algorithm. We describe other applications, such as the computation of Hilbert series of normal semigroup rings, and we indicate further connections to integer programming and statistics.