Gröbner Basis over Semigroup Algebras: Algorithms and Applications for Sparse Polynomial Systems

International audienceGröbner bases is one the most powerful tools in algorithmic non-linear algebra. Their computation is an intrinsically hard problem with a complexity at least single exponential in the number of variables. However, in most of the cases, the polynomial systems coming from applications have some kind of structure. For example , several problems in computer-aided design, robotics, vision, biology , kinematics, cryptography, and optimization involve sparse systems where the input polynomials have a few non-zero terms. Our approach to exploit sparsity is to embed the systems in a semigroup algebra and to compute Gröbner bases over this algebra. Up to now, the algorithms that follow this approach benefit from the sparsity only in the case where all the polynomials have the same sparsity structure, that is the same Newton polytope. We introduce the first algorithm that overcomes this restriction. Under regularity assumptions, it performs no redundant computations. Further, we extend this algorithm to compute Gröbner basis in the standard algebra and solve sparse polynomials systems over the torus $(C^*)^n$. The complexity of the algorithm depends on the Newton polytopes

Toric Eigenvalue Methods for Solving Sparse Polynomial Systems

We consider the problem of computing homogeneous coordinates of points in a zero-dimensional subscheme of a compact toric variety $X$. Our starting point is a homogeneous ideal $I$ in the Cox ring of $X$, which gives a global description of this subscheme. It was recently shown that eigenvalue methods for solving this problem lead to robust numerical algorithms for solving (nearly) degenerate sparse polynomial systems. In this work, we give a first description of this strategy for non-reduced, zero-dimensional subschemes of $X$. That is, we allow isolated points with arbitrary multiplicities. Additionally, we investigate the regularity of $I$ to provide the first universal complexity bounds for the approach, as well as sharper bounds for weighted homogeneous, multihomogeneous and unmixed sparse systems, among others. We disprove a recent conjecture regarding the regularity and prove an alternative version. Our contributions are illustrated by several examples.Comment: 41 pages, 7 figure

Solving a sparse system using linear algebra

We give a new theoretical tool to solve sparse systems with finitely many solutions. It is based on toric varieties and basic linear algebra; eigenvalues, eigenvectors and coefficient matrices. We adapt Eigenvalue theorem and Eigenvector theorem to work with a canonical rectangular matrix (the first Koszul map) and prove that these new theorems serve to solve overdetermined sparse systems and to count the expected number of solutions.Fil: Massri, Cesar Dario. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santalo". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santalo"; Argentin