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A Lecture on the Complexity of Factoring Polynomials over Global Fields

This paper provides an overview on existing algorithms for factoring polynomials over global fields with their complexity analysis from our experiments on the subject. It relies on our studies of the complexity of factoring parametric multivariate polynomials that is used for solving parametric polynomial systems in our PhD thesis. It is intended to be useful to two groups of people: those who wish to know what work has been done and those who would like to do work in the field. It contains an extensive bibliography to assist readers in exploring the field in more depth. The paper presents different methods and techniques used for computing irreducible factors of polynomials depending on the global field: Q, finite fields or algebraic extensions. We describe also our algorithm for factoring parametric multivariate polynomials. Mathematics Subject Classification (2000): 11Y16, 68W30, 11C08, 12D05