# Gr{\"o}bner Basis over Semigroup Algebras: Algorithms and Applications for Sparse Polynomial Systems

## TL;DR

This paper introduces a novel algorithm for computing Gröbner bases over semigroup algebras that effectively exploits sparsity in polynomial systems with varying Newton polytopes, improving efficiency in solving structured sparse systems.

## Contribution

The paper presents the first algorithm that handles diverse sparsity structures in polynomial systems over semigroup algebras, extending previous methods.

## Key findings

- Algorithm performs no redundant computations under regularity assumptions.
- Complexity depends on Newton polytopes.
- Successfully solves sparse polynomial systems over the torus.

## Abstract

Gr{\"o}bner 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 Gr{\"o}bner 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 Gr{\"o}bner 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.