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. http://dx.doi.org/10.5209/rev_REMA.1997.v10.17349 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 6k(1) 2 loseves-y integer k, tite. Sturm-Habicitt sequence associated to F is defined as tite list of polynomials {Stliaj(P)}j0,...,,, where StHa~(F) = F, StHa.,»a(P) = F’ and for everij j E {0,. .. ,p — StHa,(P) 6p—j—íSres~(P, p1⁄4 mitere Sresj(F,P’) denotes tite subresultant of indez j for P and F’. Por every j in {0 p} tite principal j-tit Stus-m—Habicitt coefficient, stha,(P), is defined as tite coefficient of x 3 itt StHa,(P). Next deflnitions introduce several sign counting functions titat we sitalí use to relate tite polynomials in tite Sturnv-Habicitt Sequence of P witit tite number of real roots of 1> itt att open interval. Definitian. Let {ao, ni,... , a,,} be a list of non zero elemente itt IP. We define: e V({a<j,ai an}) as tite nuinber of sign variatione in tite list {ao, a 1 an}, titat is tite number of consecutive sigus {+, —} or Multivariative Sturm-Habicht sequences... 121 • P({ao, a1,... , a,.}) as tite number of sign permanences in tite list {ao, a1,... ,a,4, titat is tite number of consecutiye signs{+, +} or Definition. Let 1’ be a polynomial itt I([x~ and a E F with F(cc) !=0. We define tite integesn~mber Wstn.(F; a) itt tite follo wing way: • we construct a list of polynomials {g<j, ... , g~} in 1< [x] abtained by deleting tite palynomials identically O from {StHaj(F)}s=o • Wstn.(F; a) is tite number of sign variations in tite list {go(a),. . . , g3(a)} using tite following rules for tite graups of 0’s: * we cauttt 1 sign yariation Lar tite groups: [—,0, +j, [+, 0,—], [+, 0,0,—] and [—,0,0, +] * we count 2 sign variations for tite groups: [+, 0,0, +] and [—,0,0, —1 Tite Sturm—Habicitt Str-ucture Titeos-em (see [GLRRa]))implies tbat it is not possible to finA more titan twa consecutive zeros in tite sequence {go(a),.. . , g,(a)} and titat tite sign sequences [+, 0, +], [—,0,—] can not appear. Definition. Let P be a polynomial itt E[x~ and a,¡3 E E witit a < ¡3. We define WstEa(F;a, ¡3) = WstH,(F;a) — Wstna(F;¡3) Next titeorem sitaws itow to use tite Sturm—Habicitt sequence of P and tite function Wstua to compute tite number of real roots of P ittside an apen interval Theorem. Let P be a polynomial in IK{x] and a,fl E E with a < /3 and P(a)P(/3) ~ O. Then: Wsffl8(P;a, ¡3) = J/j1{-y E (a,/3) : FQy) = 0}) 122 Laureano GonzálezVega and Guadalupe Trujillo Titis section is finisited showing itow ta use Sturm—Habicht sequence to compute tite total number of real roots (in Fi) of a polynomial in 1< [x]. First tite definition of a new sign counting functian is introduced. Definition. Let aa, ai,..., a,. be elements itt Fi with ao # O and we suppose titat .we itave tite following distribution of zeros: ={ao,.. ., a~1, O,...,0, aii+ki+1 le, aj,,0,..., 0, ai,+?e,+í,.. .,0j3,O .. , 0, at~.,+k~ ,+1, . . . , wit es-e alt tite a1 ‘s titat itave been written are notO. We define io+ko+1 = O and: 1 O({ao, ai a,.}) = Z(P({ai,~l±k. ,~1, . . . , a1)) 81 1—1 a18 })) + >j3 ~a s=1 witere: Na Cip+k.+I if = ~ (—1)2 sign( ) ~ k3 is odd ~ even Theerem. 1FF Ls a polynomial in E [x] with p = deg(F) then: C({sth4(F),. . . ,sthao(F)}) = #({‘r EF : PQy) = 0}) Volume Functions and Real Root Counting Let 11< C Fi C L be a field extension witit 1< ordered, ¡E real closed and T.~ algebraically clased. If J la a zero dimensional ideal in E W = E[xi,...,x,.] and VL(J) = {A1, ...,A8} is tite set of zeroes mt” ofj, tite main questions to be considered in this section are tite computation of tite number of A1>s itt Fi” aud tite number of A¿’s•inside a prescribed Multivariative Sturm-Habicbt sequences... 123 ii-dimensional rectangle in Fi”. Tite main toal to salve titese twa Real Root Ceunting Problems wiIl be tite Volume Funetion whicit is presented in tite next definition. Tite term “Valume Function” was introduced by P. Milne in [MiSe] tu order to compute tite number of real solutions of J inside a prescribed ii-dimensional rectangle. Deflnitian. Let¿ beapolynomialinE[~,y] = X<Ixi,...,xn,yi,...,yn]. Tite volume function associated to £ and J is the polynomial iii E [U, y ] defined by tite following equality: V¿,j(U, y) = fl (U — «t, ~» A~VL(J) witere, itt tite previous product, tite multiplicities as-e taken íttto account. Clearly, if D is tite dñnension of E W/J as E-vector space (ie tite suni of tite A¿’s multiplicities) titen tite degree of Ve,j, as polynomial in U, is equal to D. Tite Volume Function V¿,j can be determined by.computing a lexicograpitic Grabner Bases of <J, U — £(~., u)> by considering any monomial ordering verifring x> y> U. Anotiter more efficient way is basal on tite using of any Grobner Bases of J to cempute tite traces of tite pawers of ¿Qe, y) (witit respect tite extension 1K C E [~j/J), Trace((«~, 1))k) = >3 ACVL(i) and, in tite application of tite Newton Identities to recover tite cóefficients of Vj,j. Next titeorem sitows itow tite Voluine Function is useful for tite Real Root Counting Problem. Theorem. Let g be a point in Fi~ and ¿ a polynomial in 1< [~, yj venfying that j~k ~ Then the nwnber of real solutions of J (solutions in Fi”) la equal to tbe number of real roots (roots in IP) of V¿,j(U,gj: #(Vr (J)> = C({stha~(V,,j(U, a))}o.cj<D) 124 Laureano González-Vega and Guadalupe Trujillo Preof. Tite proof is very easy since tite condition impased to ¿ aud a allows te assure titat there are no salutions of J, A, in ]L” — Fi~ making £(A,a) an element of Fi: C({stha1(V,,J(U,4))}o=~=n) = #({fl E Fi : V~,jQ3,a) = 0}) = = #({/3 E Fi : Bi /3 = «A~,93}) = #(VF(J)) + +#({A~ E 12— Fi»: £(A~,a) E Fi}) = #(VF(J)) as we wanted to shaw.