Solving systems of algebraic equations

Authors:
Daniel Lazard
Affiliations:
Université de Poitiers, 86022 Poitiers Cedex, France and LIP6, Université Paris VI, 75252 Paris Cedex 05, France
Venue:
ACM SIGSAM Bulletin
Year:
2001

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Abstract

Let f1,…,fk be k multivariate polynomials which have a finite number of common zeros in the algebraic closure of the ground field, counting the common zeros at infinity. An algorithm is given and proved which reduces the computations of these zeros to the resolution of a single univariate equation whose degree is the number of common zeros. This algorithm gives the whole algebraic and geometric structure of the set of zeros (multiplicities, conjugate zeros,…). When all the polynomials have the same degree, the complexity of this algorithm is polynomial relative to the generic number of solutions.