Complexity of the resolution of parametric systems of polynomial equations and inequations

Guillaume Moroz
2006 Proceedings of the 2006 international symposium on Symbolic and algebraic computation - ISSAC '06  
Consider a parametric system of n polynomial equations and r polynomial inequations in n unknowns and s parameters, with rational coefficients. A recurrent problem is to determine some open set in the parameter space where the considered parametric system admits a constant number of real solutions. Following the works of Lazard and Rouillier, this can be done by the computation of a discriminant variety. Let d bound the degree of the input's polynomials, and σ bound the bit-size of their
more » ... ients. Based on some usual assumptions for the applications we prove that the degree of the computed minimal discriminant variety is bounded by D := (n+r)d (n+1) . Moreover we provide in this case a deterministic method which computes the minimal discriminant variety in σ O(1) D O(n+s) bit-operations on a deterministic Turing machine.
doi:10.1145/1145768.1145810 dblp:conf/issac/Moroz06 fatcat:ezstzomiunddvf7nhkcxprrajm