Multivariate Sturm—Habicht sequences: real root counting on n-rectangles and triangles

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 of suchpolynornial systems of equations inside n—dimensional rectangles ar triangles in the plane are presented. Sturm—Habicht Sequence is ane 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 of titis 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. ‘Partialí>’ supported by PRISCO (European Union, LTR 21.024) and DOES PB95-0563 (Sistemas de Ecuaciones Algebraicas: Resolución y Aplicaciones). Mathematics Subject Classification 12D10-13P05 Servicio Publicaciones Univ. Complutense. Madrid, 1997. 120 Laureano González-Vega and Guadalupe Trujillo Tite paper is divided in twa sections. In tite flrst one, tite deflnitions and main properties of Sturm—Hahicht Sequence are sitowed and tite second one is devated ta present tite notion of Multivariate Sturm— Habicitt Sequence and to present itow it can be used ta deal witit tite Real Root Counting Problem. Tite main toal to acitieve titis goal is tite generalization of tite “Volume Function” introduced in [Mime] witich is based in tite early wark of O. Hermite 011 titis tapic (see [Hermite] and [KN]). Similar farmulae to tite ones tobe presented in tite second section were aleo obtained in [Pedersen). Sturm—Habicht Sequence Let 1< be an ordered fleld and F a real-closed fleid with 11< C F. Titis section is devated ta introduce tite main properties of Sturm-.Habicitt Sequence to be used in witat follows. Tite proof of tite titeorems quoted in this section, related ta praperties of Sturm—Hahicitt sequence, can he found tu [GLRR1,2,a]. Definition. Let F be a polynomial itt E[x] with p = deg(P). If me imite >c<kl-1