Conquering inseparability: Primary decomposition and multivariate factorization over algebraic function fields of positive characteristic

  • Authors:

  • Allan Steel

  • Affiliations:

  • School of Mathematics and Statistics F07, University of Sydney, NSW 2006, Australia

  • Venue:
  • Journal of Symbolic Computation
  • Year:
  • 2005

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Abstract

Algebraic function fields of positive characteristic are non-perfect fields, and many standard algorithms for solving some fundamental problems in commutative algebra simply do not work over these fields. This paper presents practical algorithms for the first time for (1) computing the primary decomposition of ideals of polynomial rings defined over such fields and (2) factoring arbitrary multivariate polynomials over such fields. Difficulties involving inseparability and the situation where the transcendence degree is greater than one are completely overcome, while the algorithms avoid explicit construction of any extension of the input base field. As a corollary, the problem of computing the primary decomposition of a positive-dimensional ideal over a finite field is also solved. The algorithms perform very effectively in an implementation within the Magma Computer Algebra System, and an analysis of their practical performance is given.