Solving Polynomial Systems Efficiently and Accurately

Bernard Mourrain (1) , Simon Telen (2) , Marc van Barel (2)

1 AROMATH - AlgebRe, geOmetrie, Modelisation et AlgoriTHmes
2 Department of Computer Science - K.U.Leuven

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. We use new characterizations of normal forms and describe accurate and efficient constructions that allow us to compute the algebra structure of R/I, and hence the solutions of I. We show how the resulting algorithms give accurate results in double precision arithmetic and compare with normal form algorithms using Groebner bases and homotopy solvers.

Domaines: Géométrie algébrique [math.AG]; Analyse numérique [math.NA]; Calcul formel [cs.SC]

Bernard Mourrain, Simon Telen, Marc van Barel. Solving Polynomial Systems Efficiently and Accurately. 2018. ⟨hal-01738695v1⟩