Short Rational Functions for Toric Algebra and Applications

Details Short Rational Functions for Toric Algebra and Applications Short Rational Functions for Toric Algebra and Applications

2003-07-26

arxiv.org/abs/math/0307350v1

Mathematics

Combinatorics

13 pages, using elsart.sty and elsart.cls

Scientific paper

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.

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