Short Rational Functions for Toric Algebra and Applications

الأصل · EN

We encode the binomials belonging to the toric ideal Iₐ associated with an integral d × n matrix A using a short sum of rational functions as introduced by Barvinok bar,newbar. Under the assumption that d,n are fixed, this representation allows us to compute the Graver basis and the reduced Gröbner basis of the ideal Iₐ, 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.