Truncated Normal Forms for Solving Polynomial Systems: Generalized and Efficient Algorithms
For researchers in computational algebraic geometry, this work provides a more flexible and efficient framework for polynomial system solving, though it is an incremental extension of existing TNF methods.
The paper generalizes truncated normal form (TNF) algorithms for solving zero-dimensional polynomial systems, introducing non-monomial basis functions that adapt to root locations and handling non-generic systems. Experiments demonstrate improved efficiency and robustness.
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.