Sparse Gr\"obner Bases: the Unmixed Case

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 Gr\"obner bases}, an analog of classical Gr\"obner 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) Gr\"obner 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 Gr\"obner 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 Fr\"oberg's conjecture which allows us to estimate the complexity of solving overdetermined sparse systems.Comment: 20 pages, Corollary 6.1 has been corrected, ISSAC 2014, Kobe : Japan (2014

Source Galerkin Calculations in Scalar Field Theory

In this paper, we extend previous work on scalar $\phi^4$ theory using the Source Galerkin method. This approach is based on finding solutions $Z[J]$ to the lattice functional equations for field theories in the presence of an external source $J$. Using polynomial expansions for the generating functional $Z$, we calculate propagators and mass-gaps for a number of systems. These calculations are straightforward to perform and are executed rapidly compared to Monte Carlo. The bulk of the computation involves a single matrix inversion. The use of polynomial expansions illustrates in a clear and simple way the ideas of the Source Galerkin method. But at the same time, this choice has serious limitations. Even after exploiting symmetries, the size of calculations become prohibitive except for small systems. The calculations in this paper were made on a workstation of modest power using a fourth order polynomial expansion for lattices of size $8^2$,$4^3$,$2^4$ in $2D$, $3D$, and $4D$. In addition, we present an alternative to the Galerkin procedure that results in sparse matrices to invert.Comment: 31 pages, latex, figures separat