Sparse Gröbner bases
Jean‐Charles Faugère, Pierre-Jean Spaenlehauer, Jules Svartz · 2014
Toric (or sparse) elimination theory is a framework developped during the last decades to exploit monomial structures in systems of Laurent polynomials. Roughly speaking, this amounts to computing in a semigroup algebra, i.e. an algebra generated by a subset of Laurent monomials. In order to solve symbolically sparse systems, we introduce sparse Gröbner bases, an analog of classical Gröbner bases for semigroup algebras, and we propose sparse variants of the F5 and FGLM algorithms to compute them. Our prototype "proof-of-concept" implementation shows large speedups (more than 100 for some examples) compared to optimized (classical) Gröbner bases software. Moreover, in the case where the generating subset of monomials corresponds to the points with integer coordinates in a normal lattice polytope P ⊂ Rn and under regularity assumptions, we prove complexity bounds which depend on the combinatorial properties of P. These bounds yield new estimates on the complexity of solving 0-dim systems where all polynomials share the same Newton polytope (unmixed case). For instance, we generalize the bound min(n1, n2) + 1 on the maximal degree in a Gröbner basis of a 0-dim. bilinear system with blocks of variables of sizes (n1, n2) to the multilinear case: Σ ni - max(ni) + 1. We also propose a variant of Fröberg's conjecture which allows us to estimate the complexity of solving overdetermined sparse systems. Finally, our complexity results apply in the dense (usual) case and, as a surprising by-product, we prove that restrictive assumptions in usual complexity estimates of classical inhomogeneous Gröbner bases algorithms can be removed.