Multivariate Sturm-Habicht sequences:realroot counting on n-rectangles and triangles

Abstract

Tite main purpose of titis note is to show how Sturm—Habicht Sequence can be generalized to the multivariate case and used to compute tbe number of real solutions of a polynamial system of equations with a finite number of complex solutions. Using tite same techniques, sorne formulae counting the number of real salutions ofsuchpolynornial systems ofequations inside n—dimensional rectangles ar triangles in the plane are presented. Sturm—Habicht Sequence isane of tite toals titat Computational Real Algebraic Geometry provides to deal witit tite prablem of computing tite number of real roots of an univariate polynomial in 7Z[x] witit goad specialization praperties and cantrolled complexity (see [GLRR,, 2,SD. Tite purpose oftitis note is to shaw itow Sturm—Habicitt Sequence can be easily generalized to tite multivariate case and used to compute tite number of real solutions of a polynamial system of equations witit a finite number of complex solutians. Using tite same tecitnics it will he sitowed itaw to count real solutians of sucit polynamials systems of equations inside n-dimensional rectangles or tu triangles in tite plane. Titese counting algorititms will work anly witen. tite considered polynomial system of equations itas a finite nuinher of complex solutions.