Efficient Algorithms and Bounds for Wu-Ritt Characteristic Sets

Abstract

The concept of a characteristic set of an ideal was originally introduced by J.F. Ritt, in the late forties, and later, independently rediscovered by Wu Wen-Tsiin, in the late seventies. Since then Wu-Ritt Characteristic Sets have found wide applications in Symbolic Computational Algebra, Automated Theorem Proving in Elementary Geometries and Computer Vision. The original algorithm of Ritt, and subsequent modifications by Wu, has a non-elementary worst-case time complexity, and could be used for computing only an extended characteristic set. In this paper, we present optimal algorithms for computing a characteristic set with simple-exponential sequential and polynomial parallel time complexities. These algorithms are derived, via linear algebra, from simple-exponential degree bounds for a characteristic set. The degree bounds are obtained by using the recent effective version of Hilbert’s Nullstellensatz, due to D. Brownawell and J. Kollár, and a version of Bezout’s Inequality, due to J. Heintz.KeywordsMonic PolynomialDifferential AlgebraParallel TimeAutomate Theorem ProveDegree BoundThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.