Sparse Gröbner Bases: the Unmixed Case

Abstract

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 \emph{semigroup algebra}, \emph{i.e.} an algebra generated by a subset of Laurent monomials. In order to solve symbolically sparse systems, we introduce \emph{sparse Gröbner bases}, an analog of classical Gröbner bases for semigroup algebras, and we propose sparse variants of the $F_5$ and FGLM algorithms to compute them. Our prototype "proof-of-concept" implementation shows large speed-ups (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 $\mathcal{P}\subset {\mathbb{R}}^n$ and under regularity assumptions, we prove complexity bounds which depend on the combinatorial properties of $\mathcal{P}$. These bounds yield new estimates on the complexity of solving $0$-dim systems where all polynomials share the same Newton polytope (\emph{unmixed case}). For instance, we generalize the bound $\min (n_1,n_2)+1$ on the maximal degree in a Gröbner basis of a $0$-dim. bilinear system with blocks of variables of sizes $(n_1,n_2)$ to the multilinear case: $\sum n_i-\max (n_i)+1$. We also propose a variant of Fröberg's conjecture which allows us to estimate the complexity of solving overdetermined sparse systems.

20 pages, Corollary 6.1 has been corrected, ISSAC 2014, Kobe : Japan (2014)

References in corpus (1)

• A survey on signature-based Gröbner basis computations

Cited by in corpus (6)

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• Towards Mixed Gr{ö}bner Basis Algorithms: the Multihomogeneous and Sparse Case
• Learning a performance metric of Buchberger's algorithm
• On the chordality of polynomial sets in triangular decomposition in top-down style
• Sparse Polynomial Systems with many Positive Solutions from Bipartite Simplicial Complexes
• Solving sparse polynomial systems using Groebner bases and resultants