Truncated Normal Forms for Solving Polynomial Systems: Generalized and Efficient Algorithms

TL;DR

This paper advances the computation of polynomial system solutions by introducing generalized truncated normal forms with special basis functions, improving efficiency and adaptability for zero-dimensional ideals.

Contribution

It introduces new basis functions for TNFs, generalizes algorithms for non-generic systems, and demonstrates improved efficiency and adaptability in solving polynomial systems.

Findings

New basis functions improve solution localization

Algorithms efficiently handle non-generic systems

Experimental results show practical effectiveness

Abstract

We consider the problem of finding the isolated common roots of a set of polynomial functions defining a zero-dimensional ideal I in a ring R of polynomials over C. Normal form algorithms provide an algebraic approach to solve this problem. The framework presented in Telen et al. (2018) uses truncated normal forms (TNFs) to compute the algebra structure of R/I and the solutions of I. This framework allows for the use of much more general bases than the standard monomials for R/I. This is exploited in this paper to introduce the use of two special (nonmonomial) types of basis functions with nice properties. This allows, for instance, to adapt the basis functions to the expected location of the roots of I. We also propose algorithms for efficient computation of TNFs and a generalization of the construction of TNFs in the case of non-generic zero-dimensional systems. The potential of the TNF method and usefulness of the new results are exposed by many experiments.