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## Section: New Results

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

Participant : Bernard Mourrain.

In [16], 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 (non-monomial) 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.

This is a joint work with Simon Telen and Marc Van Barel, Department of Computer Science - K.U.Leuven.