INTRODUCTION Introduction The core of this paper consists in the exhibition of a system of canonical equations which describe for generic coordinates

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1 Equations for Polar Varieties and Ecient Real Elimination B Bank, M Giusti 3, J Heintz 4, G M Mbakop Dedicated to Steve Smale January 6, 000 Abstract Let V 0 be a smooth and compact real variety given by a reduced regular sequence of polynomials f ::: f p This paper is devoted to the algorithmic problem of nding eciently for each connected component of V 0 a representative point For this purpose we exhibit explicit polynomial equations which describe for generic variables the polar varieties of V 0 of all dimensions This leads to a procedure which solves our algorithmic problem in time that is polynomial in the (extrinsic) description length of the input equations f ::: f p and in a suitably introduced geometric (extrinsic) parameter, called the degree of the real interpretation of the given equation system f ::: f p Keywords: Real polynomial equation solving, polar variety, geometric degree, arithmetic circuit, arithmetic network, complexity Research partially supported by the following German, French, Spanish and Argentinian grants: BA 57/4- (DFG), ARG 08/98 INF (BMBF), ECOS A99E06, DGI- CYT PB C0-0, ANPCyT , UBACYT TW 80 and PIP CONICET 457/96 The rst two authors wish to thank the MSRI at Berkeley for its hospitality during their stay fall 998 Humboldt-Universitat zu Berlin, Inst fur Mathematik, 0099 Berlin, Germany bank@mathematikhu-berlinde, mbakop@mathematikhu-berlinde 3 Laboratoire GAGE, Ecole Polytechnique, 98 Palaiseau Cedex, France giusti@gagepolytechniquefr 4 Dept de Matematicas, Estadstica y Computacion, Facultad de Ciencias, Universidad de Cantabria, 3907 Santander, Spain heintz@matescounicanes and Dept de Matematica, Universidad de Buenos Aires, Ciudad Univ, PabI, 48 Buenos Aires, Argentina joos@matedmubaar

2 INTRODUCTION Introduction The core of this paper consists in the exhibition of a system of canonical equations which describe for generic coordinates (locally) the polar varieties of a given semialgebraic complete intersection manifold V 0 contained in the real n {dimensional ane space IR n This (purely mathematical) description of the polar varieties allows the design of a new type of ecient algorithm (with intrinsic complexity bounds), which computes, in case that V 0 is smooth and compact, at least one representative point for each connected component of V 0 (the algorithm returns each such point in a suitable symbolic codication) This new algorithm (and, in particular, its complexity) is the main practical outcome of the present paper Let us now describe briey our results Suppose that the semialgebraic variety V 0 is compact and given by polynomial equations of the following form: f (X ::: X n )= = f p (X ::: X n )=0 where p n IN p n and f ::: f p belong to the polynomial ring Q [X ::: X n ] in the indeterminates X ::: X n over the rational numbers Q Let d be a given natural number and assume that for k p the total degree deg f k of the polynomial f k is bounded by d Moreover, we suppose that the polynomials f ::: f p form a regular sequence in Q [X ::: X n ] and that they are given by a division-free arithmetic circuit of size L that evaluates them in any given point of the real (or complex) n {dimensional ane space IR n (or C n ) Further, we assume that the Jacobian J(f ::: f p ) of the equation system f = = f p =0 has maximal rank in any point of V 0 (thus, implicitly we assume that V 0 is smooth) Let W 0 := V (f ::: f p ) denote the (complex) algebraic variety dened by the polynomials f ::: f p in the ane space C n We denote the singular locus of W 0 by SingW 0 Moreover, let us suppose that the variables X ::: X n are in generic position with respect to the equation system f ::: f p For i n ; p let W i be the i {th formal (complex) polar variety associated with W 0 (and the variables X p+i ::: X n ) Further, let us denote the real counterpart of W i by V i := W i \ IR n We call V i the i {th formal real polar variety associated with the real semialgebraic variety V 0 (and the variables X p+i ::: X n ) It turns out that the (locally) closed sets W i n SingW 0 and V i are either empty or complex or real manifolds of dimension n ; (p + i) Moreover, for i n ; p, one sees easily that fw i := W i n SingW 0 is the i {th polar variety (in the usual sense) associated with W 0 and the variables X p+i ::: X n (here, W i n SingW 0 denotes the Q {Zariski closure

3 INTRODUCTION 3 in C n of the algebraic set W i n SingW 0 ) For a precise denition of the notion of formal polar varieties and of polar varieties in the usual sense we refer to Section Suppose that the real variety V 0 is non{empty and satises our assumptions In Theorem?? of this paper we show that every real polar variety V i = W i \ IR n i n ; p is a non{empty, smooth manifold of dimension n ; p ; i containing at least one point ofeach connected component ofthe real variety V 0 In particular, the real variety V n;p is a nite set, containing at least one representative point ofeach connected component of V 0 Under the same assumptions weshow in Theorem?? that, for i n;p, the algebraic set W i n SingW 0 can be described locally by complete intersection ideals that satisfy the Jacobian criterion More precisely, the algebraic set W i n SingW 0 is a smooth manifold of codimension p + i that can be described locally by certain regular sequences consisting of the polynomials f ::: f p and i many well{determined p {minors of the Jacobian J(f ::: f p ) of the f ::: f p In particular, the algebraic set W n;p nsingw 0 is zero-dimensional, whence Wn;p f = W n;p nsingw 0 Thus fw n;p is a zero-dimensional complex variety that contains a representative point of each connected component of the real variety V 0 The practical outcome of Theorem?? and Theorem?? consists in the design of an ecient algorithm (with intrinsic complexity bounds), which adapts the elimination procedure for complex algebraic varieties developed in [?] and [?] to the real case Under the additional assumption that, for k p, the intermediate ideal (f ::: f k ) generated by f ::: f k in Q [X ::: X n ] is radical, we shall apply this procedure to the p ; n p; well{determined equation systems of Theorem??, which describe locally the zero-dimensional algebraic variety Wn;p f = W n;p n SingW 0 In order to nd at least one representative pointforevery connected component ofthe real variety V 0,wehave just to run the procedure of [?] and [?] on all these equation systems Counting arithmetic operations in Q at unit costs, this can be done in sequential time n p ;! L(nd) O() where is the following geometric (and therefore, intrinsic) invariant: := maxfmaxfdeg V (f ::: f k ) n SingW 0 j k pg maxfdeg f Wi j i n ; pgg (here, deg V (f ::: f k ) n W 0 and deg f Wi denote the geometric degree in the sense of [?] of the corresponding algebraic varieties)

4 INTRODUCTION 4 This is the content of Theorem?? below The quantity bounds for any k p and any i n ; p the degree of the algebraic variety V (f ::: f k ) n SingW 0 and of the i {th polar variety f Wi = W i n SingW 0 In [?] and [?] the quantity maxfdeg V (f ::: f i )j i pg is called the geometric degree (of the complex interpretation) of the equation system f ::: f p In analogy to this terminology, we shall call the geometric degree of the real interpretation of the equation system f ::: f p In view of the complexityresultabove we shall understand the parameter as an intrinsic measure for the size of the real interpretation of the given polynomial equation system In order to make our complexity result more transparent we are going now to exhibit, in terms of extrinsic parameters, some estimations for the intrinsic system degree Let us write d := deg f ::: d p := deg f p and let us denote by D := d d p the classical Bezout number of the polynomial system f ::: f p Then we have the following degree estimations for the complex algebraic variety W 0 = V (f ::: f p ) deg V 0 deg W 0 D d p ( V 0 denotes again the Q {Zariski{closure in C n of the real variety V 0 ) On the other hand, we conclude from Theorem?? that, for every i i n ; p the polar variety Wi f is dened by theinitial system f ::: f p and certain p {minors of the Jacobian J(f ::: f p ) Let us denote by c i the maximum degree of these p {minors It turns out that for any i n ; p the polar variety Wi f is a codimension one subvariety of Wi; f Now one sees easily that the quantity D i := D c c i represents a reasonable "Bezout number" of the variety Wi f and that this Bezout number satises the estimate deg Wi f D i Putting all this together, we deduce the following estimate for the intrinsic system degree : D n;p = Dc c n;p : Observing that for any i i n;p the inequality c i d ++ d p ; p holds, we nd the estimations: D(d + + d p ; p) n;p d p (pd ; p) n;p <p n;p d p (d ; ) n;p <p n;p d n : In conclusion, our new real algorithm has a time complexity that is, in worst case, polynomial in the "Bezout number" Dc c n;p of the zero{ dimensional polar variety Wn;p f Our complexity bound ; n p; L(nd) O() depends in a polynomial manner on the intrinsic (geometric, semantic) parameter and on the extrinsic (algebraic) parameters d and n, and it depends only linearly on the syntactic

5 POLAR VARIETIES 5 parameter L In this sense one may consider our complexity bound as intrinsic Our real algorithm promises therefore to be practically applicable to special equation systems with low value for the intrinsic parameter On the other hand, also in worst case our algorithm improves upon the known d O(n) {time procedures for the algorithmic problem under consideration, even in their most ecient versions [?], [?] (see also [?], [?], [?], [?], [?], [?], [?], [?], [?], [?]) However this distinction does not become apparent when we measure complexities simply in terms of d and n (all mentioned algorithms have worst{case complexities of type d O(n) ), but it becomes clearly visible when we use the "Bezout number" just introduced as complexity parameter Only our new algorithm is polynomial in this quantity Our (algorithmic and mathematical) methods and results represent a non{ obvious generalization of the main outcome of [?], where an intrinsic type algorithm was designed for the problem of nding at least one representative point in each connected component of a real, compact hypersurface given by an n {variate, smooth polynomial equation f of degree d with rational coecients (such that f represents a regular equation of that hypersurface) This is the particular case of codimension p = of the present paper, and our setting leads to the complexity bound L(nd) O() proved in [?] Polar Varieties Notations, Notions and General Assumptions Let X ::: X n be indeterminates (variables) over the rational numbers Q and let be given polynomials f ::: f p Q [X ::: X n ] with p n Let C n and IR n denote the n {dimensional ane space over the complex and the real numbers, respectively We think C n to be equipped with the Q {Zariski topology, whereas on IR n we consider the strong (euclidian) topology For any subset U C n we deote by U its Q {Zariski{closure By X := (X ::: X n ) we denote the vector of variables X ::: X n and by x := (x ::: x n ) any point of the ane space C n or IR n We suppose that the polynomials f ::: f p form a reduced regular sequence in Q [X ::: X n ] The Jacobian of these polynomials is denoted by " J(f ::: f p ):= j kp jn For any point x C n we write J(f ::: f p )(x) := " j kp jn

6 POLAR VARIETIES 6 for the Jacobian of the polynomials f ::: f p at x The common complex zeroes of the polynomials f ::: f p Q {denable subvariety of C n which we denote by form an ane, W 0 := V (f ::: f p ):=fx C n jf (x) =::: = f p (x) =0g: A point x W 0 = V (f ::: f p ) is said to be smooth (in W 0 ) if the rank of the Jacobian of f ::: f p in x satises the condition rk J(f ::: f p )(x) = p Otherwise x is called a singular point of W 0 By SingW 0 we denote the set of all singular points of W 0 Remark If x W 0 is smooth, then the hypersurfaces dened by the polynomials f ::: f p intersect transversally at the point x Denition For every i i n ; p let i denote the set of all common complex zeros of all p -minors of the Jacobian J(f ::: f p ) corresponding to the columns f ::: p+ i ; g In other words, i is the determinantal variety dened by all p-minors of the submatrix J p+i; (f ::: f p ) determined by the columns f ::: p+ i ; g of the Jacobian J(f ::: f p ) We introduce the ane variety W i := W 0 \ i associated with the linear subspace of C n, namely X p+i; := fx C n jx p+i (x) =:::= X n (x) =0g and call W i the i {th formal polar variety of W 0 By fw i := W i n SingW 0 we denote the i {th polar variety (in the usual sense) of the variety W 0 eg [?]) (see Remark 3 The index i reects the expected codimension of the formal polar variety W i in W 0 With respect to the ambient space C n, one has always codim W i p + i: According to our notation, the common zeros of all p {minors of the Jacobian J(f ::: f p ) form the determinantal variety n;p+ Obviously, wehave Sing W 0 = W 0 \ n;p+ = W n;p+ The formal polar varieties W i i n ; p form a decreasing sequence In particular, we have W 0 W W i W n;p W n;p+ = Sing W 0

7 POLAR VARIETIES 7 Local Description of the Determinantal Varieties In this subsection we develop a succinct local description of the determinantal varieties i i n ; p The following general Exchange Lemma will be our main tool for this description (this lemma is used in a similar form in [?]) It describes an exchange relation between certain minors of a given matrix Let A be a given (p n) -matrix with entries a ij from an arbitrary commutative ring Let l and k be any natural numbers with l n and k minfp lg Furthermore, let I k := (i ::: i k ) be an ordered sequence of k dierent elements from the nite set of natural numbers f ::: lg and let M A (I k ):=M A (i ::: i k ) denote the k -minor of the matrix A built up by the rst k rows and the columns i ::: i k If it is clear by the context which is the matrix A,we shall just write M(i ::: i k ):=M(I k ):=M A (I k ) Lemma 4 (Exchange Lemma) Let be given as before a matrix A and natural numbers l and k, and let be given two intersecting index sets I k =(i ::: i k ) and I k; =(j ::: j k; ) Then, for suitable numbers " j f ;g with j I k n I k; we have the following identity: X () M (I k; ) M (I k )= " j M (I k nfjg) M (I k; [fjg) : ji k ni k; Proof Consider the following ((k ; ) (k ; )) -matrix L with entries from the given matrix A : L := 6 4 O L (I k ) L k; (I k ) L (I k; ) L (I k ) L k (I k; ) L k (I k ) : Here, for any j k L j (I k ) denotes the row vector of length k that we obtain selecting from the j {th row of the matrix A the k elements placed in the columns I k = (i ::: i k ) Similarly, L j (I k; ) is obtained from the j {th row of A selecting the k ; elements placed in the columns I k; =(j ::: j k; ) It is now not dicult to verify the identity () by calculating the determinant det L of the quadratic matrix L via Laplace expansion in two dierent

8 POLAR VARIETIES 8 ways First, by expansion of det L according to the rst k ; columns of L,we obtain the left{hand side of () disregarding the sign Expansion of det L according to the rst k ; rows of L leads to the right{hand side of () This implies the identity () for an appropriate choice of the signs " j, with j I k n I k; Let m Q[X ::: X n ] denote the (p;) {minor of the Jacobian J(f ::: f p ) given by the rst (p ; ) rows and columns, ie, let " m := det j kp; jp; We consider the determinantal variety i V (m) :=fx C n j m(x) =0g outside of the hypersurface and denote this localization by ( i ) m, ie, we set ( i ) m := i n V (m): From now on,for i i p n, let us denote by M(i ::: i p ) the polynomial in Q [X ::: X n ] dened as the p {minor of the Jacobian J(f ::: f p ) built up by its p rows and the columns i ::: i p As before, we denote by M(i ::: i p )(x) the specialization of M(i ::: i p ) in a given point x C n Proposition 5 Let i n ; p be arbitrarily xed, and let m be the (p ; ) { minor dened above Then the determinantal variety i is locally (ie, outside of the hypersurface V (m) ) described by the i polynomials M( ::: p; p) M( ::: p; p+) ::: M( ::: p; p+ i ; ): In other words, we have ( i ) m := fx C n j m(x) 6= 0 M( ::: p; s)(x) =0 sfp : : : p+i;gg where M( ::: p; s) denotes, as above, the p {minor of the Jacobian J(f ::: f p ) built up by the rst p ; columns and the s {th column

9 POLAR VARIETIES 9 Proof It suces to show that ( i ) m fx C n j m(x) 6= 0 M( ::: p; s)=0 sfp : : : p + i ; gg holds Let x C n be anypoint satisfying the conditions m(x ) 6= 0and M( ::: p; s)(x )=0 for every s fp : : : p + i ; g We have toverify that M(i ::: i p )(x )=0 holds for all ordered p {tuples (i ::: i p ) of elements of f ::: p+ i ; g Applying the Exchange Lemma to m = M( ::: p; ) and M(i ::: i p ), we deduce the identity = X jfi ::: i pgnf ::: p;g m(x )M(i ::: i p )(x )= " j M (fi ::: i p gnfjg)(x )M( ::: p; j)(x ) for suitable numbers " j f ;g with j fi ::: i p gnf ::: p; g By assumption we have m(x ) 6= 0 and M( ::: p; j)(x ) = 0 for all j fp : : : p + i ; g This implies that x belongs to the set ( i ) m Notation 6 In the sequel we shall just write M j for the p { minor M( ::: p; j) given by the rst p ; columns of J(f ::: f p ) and the column j fp ::: ng outside of the hy- Remark 7 Proposition?? implies that the codimension of i persurface V (m) is at most i Proposition?? holds also for the determinantal variety n;p+ that denes the singular locus Sing W 0 = W n;p+ of the variety W 0 Hence, for any point x C n satisfying the condition m(x ) 6= 0 and the n ; p + equations M j (x )=0 j fp : : : ng the Jacobian J(f ::: f p )(x ) becomes singular Replacing the previously chosen (p;) -minor m by any other (p;) - minor of the Jacobian J(f ::: f p ), the statement of Proposition?? remains true mutatis mutandis

10 POLAR VARIETIES 0 3 Local Description of the Formal Polar Varieties The aim of this subsection is to show the following fact: Let the variables X ::: X n be in generic position with respect to the polynomials f ::: f p, and let ~m be any (p ; ) {minor of the Jacobian J(f ::: f p ) In this subsection we are going to show that any formal polar variety W i i n ; p is a smooth complete intersection variety outside of the closed set SingW 0 [ V (~m) Moreover, we shall exhibit a reduced regular sequence describing this variety outside of SingW 0 [ V (~m) As in the previous subsection, let m Q [X ::: X n ] denote the (p ; ) { minor of the Jacobian J(f ::: f p ) built up by the rst (p ; ) rows and columns Let Y ::: Y n be new variables and let Y := (Y ::: Y n ) For any linear coordinate transformation X = AY, with A being a regular (n n){ matrix, we dene the polynomials G (Y ):=f (AY ) ::: G p (Y ):=f p (AY ): The Jacobian of G ::: G p has the form J(G ::: G p j # kp jn Using a similar notation as before, we denote by fm(i ::: i p ) = J(f ::: f p )A: the p {minor of the new Jacobian J(G ::: G p ) that corresponds to the columns i < <i p n Moreover, we denote by Mj f the p { minor M( ::: p; f j) determined by the xed rst p ; columns of J(G ::: G p ) and the column j fp : : : ng For p r t n let Z r t be a new indeterminate Using the following regular (n ; p +) (n ; p + ) {parameter matrix Z := Z p+ p Z p+i; p Z p+i; p+ Z p+i p Z p+i p+ Z p+i p+i; O 0 Z n p Z n p+ Z n p+i; Z n p+i Z n p+i

11 POLAR VARIETIES we construct an (n n) {coordinate transformation matrix A := A(Z) that will enable us to prove the statement at the beginning of this subsection For the moment,letusxanindex i n ; p We consider the formal polar variety W i outside of the hypersurface V (m) Corresponding to our choice of i, the matrix Z may be subdivided into submatrices as follows: 6 Z = 4 Z (i) O i n;p;i+ Z (i) Z (i) : Here the matrix Z (i) is dened as Z (i) := 6 4 Z p+i p ::: Z p+i p+i; ::: ::: ::: Z np ::: Z n p+i; and Z (i) and Z (i) denote the quadratic lower triangular matrices bordering Z (i) in Z,and O i n;p;i+ is the i (n ; p ; i +) zero matrix Let A := A(Z) := 6 4 I p; O p; i O p; n;p;i+ O i p; Z (i) O i n;p;i+ O n;p;i+ p; Z (i) Z (i) Here the submatrices I r and O r s are unit or zero matrices, respectively, of corresponding size, and Z (i) Z (i) : and Z (i) are the submatrices of the parameter matrix Z introduced before Thus, A is a regular, parameter dependent (n n) {coordinate transformation matrix Like the matrix Z, the matrix A contains s := (n ; p)(n ; p +) parameters Z r t whichwemay specialize into any point z of the ane space C s For suchapoint z C s we denote by A(z), Z (i) (i) (z) Z (z) and Z (i) (z) for the corresponding specialized matrices We consider now the coordinate transformation given by X = AY with A = A(Z) and calculate the Jacobian J(G ::: G p ) with respect to the new polynomials G ::: G p Recall that the coordinate transformation matrix A depends on our previous choice of the index i n ; p According to the structure of the coordinate transformation matrix A = A(Z) we subdivide the Jacobian J(f ::: f p ) into three submatrices

12 POLAR VARIETIES J(f ::: f p )= U V W with U := j kp jp; V := j kp pjp+i; W := j kp p+ijn From the identity J(G ::: G p )=J(f ::: f p ) A we deduce that our new Jacobian is of the form: " J(G ::: G p )= = U VZ + WZ (i) WZ (i) : j kp jn We are interested in a local description of the i {th formal polar variety W i = W 0 \ i outside of the hypersurface V (m), where m is the xed upper left (p ; ) { minor of the Jacobian J(f ::: f p ) (and also of its submatrix U ) Since the coordinate transformation X = AY leaves the submatrix U unchanged, the p; {minor m remains xed under this transformation From Proposition??, weknow that the localized determinantal variety ( i ) m is described by the i equations M p =0 ::: M p+i; =0 and by the condition m 6= 0 The p { minors M p ::: M p+i; dening these equations are built up by the submatrix [ U V ] of the Jacobian J(f ::: f p ) Under the coordinate transformation A(Z) the matrix [ U V ] is changed into the submatrix h U VZ (i) + WZ (i) i of the Jacobian J(G ::: G p ) and the p {minors M p ::: M p+i; are changed into the p { minors fm p ::: Mp+i; f h i of the matrix U VZ (i) + WZ (i) This implies the matrix identity : () h fmp ::: f Mp+i; i =[M p ::: M p+i; ] Z (i) +[M p+i ::: M n ] Z (i) : For the previously chosen index i n;p, the coordinate transformation X = A(Z)Y induces the following morphism of ane spaces: i : C n C s! C p C i

13 POLAR VARIETIES 3 dened by (x z) 7;! i (x z) := f (x) ::: f p (x) Mp f (x z) ::: Mp+i; f (x z) Consider an arbitrary point z C s We denote by h z i the determinantal subvarietyof C n dened byall p {minors of the matrix U VZ (i) (z)+wz(i) (z) (which is a submatrix of the new Jacobian obtained by specializing the coecients of the polynomials G ::: G p into the point z C s ) Writing Wi z := W 0 \ z i, one sees immediately that the zero ber ; i (0) of the morphism i contains the set (W z i ) m := W 0 \ ( z i ) m : In other words, for any arbitrarily chosen point z C s, the zero ber ; i (0) of the morphism i contains the transformed formal polar variety Wi z, localized in the hypersurface V (m) and expressed in the old coordinates We are going now to analyze the rank of the Jacobian of the morphism i in an arbitrary point (x z) C n C s with x (Wi z) m Using the subdivision of the parameter matrix Z into the parts Z (i) Z (i) and Z (i), the Jacobian J( i ) of the morphism i can be written symbolically as " i J( i )= We have = @Z i (i) J(f ::: f p ) O p n;p;i+ O p e p+i p e np O n;p;i+ O n;p;i+ Mp+i; e Mp+i; p+i n p+i; where the columns correspond to the partial derivatives of i with respect to the variables i X ::: X n Z p+i p ::: Z n p ::: Z p+i p+i; ::: Z n p+i; (in this order) The entries O r t denote here zero matrices of corresponding size and the row matrices labeled by " " represent the partial derivatives with respect to the variables X ::: X n of the minors Mp f ::: f Mp+i; This row matrices will be irrelevant for our considerations

14 POLAR VARIETIES 4 Furthermore, the third submatrix 6 e p+ (i) # of J( i ) canbewrittenas O p i; O p i; 0 ::: O e p+i; p O i; Mp+ Mp e p+ p+i; p+ O i; O i; e p+i; p+i; O i; O i; 0 and the last (i) # of J( i ) is a zero matrix since the p { minors Mp f ::: f Mp+i; are indepedent ofthe parameters Z r t occuring in the submatrix Z (i) of the coordinate transformation matrix A(Z) Therefore, the Jacobian J( i ) is of full rank p + i wherever the submatrix ej( @ (i) is of full rank p + i On the other hand, considering for p j p + i ; the in e J(i ) contained i row matrices Mj f f # p+i n j we see that the representation () of the transformed p {minors Mj f implies the identity Mj f f # Mj =[M p+i ::: M n ] p+i n j Thus, we obtain the representation ej( i )= 6 4 J(f ::: f p ) O p n;p;i+ O p n;p;i+ [M p+i ::: M n ] O n;p;i+ O n;p;i+ [M p+i ::: M n ] Since all entries of the submatrix J(i e ) of the Jacobian J( i ) belong to the polynomial ring Q [X ::: X n ], we see that the rank of the matrix 3 : 7 5

15 POLAR VARIETIES 5 J( i ) in a given point (x z) C n C s with x (Wi z) m depends only on the choice of x According to our localization outside of the hypersurface V (m), let us consider an arbitrary smooth point ~x of W 0 = V (f ::: f p ) satisfying the condition m(~x) 6= 0 Suppose that the submatrix J(i e )(~x) is not of full rank, ie, that rk e J(i )(~x) <p+ i holds This latter inequalityisvalid if and only if all p -minors M p+i ::: M n of the Jacobian J(f ::: f p ) vanish at ~x Let ~z C s be any parameter point such that the pair (~x ~z) belongs to the ber ; i (0) of the morphism i Since the p -minors Mp ~ ::: Mp+i; ~ of the transformed Jacobian J(G ::: G p ) must vanish at (~x ~z) we deduce from () that [0 ::: 0] = [M p (~x) ::: M p+i; (~x)] Z (i) (~z) holds (here Z (i) (~z) denotes again the matrix obtained by specializying the entries of Z (i) into the corresponding coordinates of the point ~z C s ) Because of the lower triangular form of the regular matrix Z (i), the latter matrix equation holds if and only if the conditions M p+i; (~x) == M p (~x) =0: are satisfyied Therefore, our assumptions on ~x and ~z imply m(~x) 6= 0 and M p (~x) = = M n (~x) = 0: However, by Remark?? this means that the Jacobian J(f ::: f p )(~x) is singular Hence, ~x is not a smooth point of W 0,ie, ~x Sing W 0, which contradicts our assumption on ~x Now, suppose that we are given a point (x z) C n C s that belongs to the ber ; i (0) Then x belongs to W 0 Further, suppose that x is a smooth point of W 0 outside of the hypersurface V (m) Let us consider the Zariski{open neighbourhood U ~ of x consisting of all points x Cn with m(x) 6= 0 and rk J(f ::: f p )=p, ie, we consider eu := C n n (Sing W 0 [ V (m)) : We are going to show that the restricted morphism i : ~ U C s! C p C i is transversal to the origin 0 C p C i In order to see this, consider an arbitrary point (x z) of U ~ Cs that satises the equation i (x z) =0 Thus, x belongs to U ~ \ W0 and is, therefore, a smooth point of W 0, which is outside of the hypersurface V (m) By the preceding considerations on the rank of the Jacobian J( i ) it is clear that J( i ) has the maximal rank p + i at (x z) This means that (x z) is a

16 POLAR VARIETIES 6 regular pointof i Since (x z) was an arbitrary pointof ; i (0)\ ( ~ U Cs ), the claimed transversality has been shown Now, applying the Weak{Transversality{Theorem of Thom{Sard (see eg [?]) to the diagram ; i (0) \ ( ~ U Cs ),! C n C s & # one concludes that there is a residual dense set i of parameters z C s for which transversality holds This implies that, for every xed z i, the transformed and localized formal polar variety W z i n (Sing W 0 [ V (m)) is either empty or a smooth variety of codimension p + i This variety can be described locally by the polynomials () f (X) ::: f p (X) f Mp (X z) ::: f Mp+i; (X z) that form outside of SingW 0 [ V (m) a regular sequence Up to now, our considerations concerned only to the change of coordinates for an arbitrarily xed i n ; p However, := T n;p i= i is a dense residual parameter set in C s from which we can choose a simultaneous change of coordinates for all i n ; p For every choice z and i n ; p the transformed formal polar variety Wi z is outside of the closed set SingW 0 [ V (m) a smooth complete intersection variety described by the (local) regular sequence ( ) One sees now easily that the ane space IR s contains a non{empty residual dense set of parameters z such that the conclusion above apply to the coordinate transformation X = A(z)Y Moreover z can chosen from Q s Taking into account Proposition?? and Remark??, we deduce from our argumentation the following result: Theorem 8 Let W 0 = V (f ::: f p ) be a reduced complete intersection variety given by polynomials f ::: f p in Q[X ::: X n ] and suppose that the variables X ::: X n are in generic position with respect to f ::: f p Further, let m be the upper left (p ; ) {minor of the Jacobian J(f ::: f p ) Then, every formal polar variety W i i n ; p localized with respect to the closed set SingW 0 [ V (m), is either empty or a smooth variety of codimension p + i that can be decribed by the equations f ::: f p M p ::: M p+i; C s

17 POLAR VARIETIES 7 where M j p j p + i ; is the p { minor of the Jacobian J(f ::: f p ) given by the columns ::: p; j The polynomials f ::: f p M p ::: M p+i; form then outside of SingW 0 [ V (m) a regular sequence Remark 9 Taking into account that the argumentation on the localization with respect to the xed (p ; ) -minor m remains valid mutatis mutandis for any other (p;) -minor ~m of the Jacobian J(f ::: f p ), Theorem?? can be restated for any xed (p ; ) {minor just by reordering of columns and rows of the Jacobian J(f ::: f p ) 4 Existence of Real Points in the Polar Varieties Let f ::: f p Q[X ::: X n ] be a reduced regular sequence and let again W 0 := V (f ::: f p ) be the ane variety dened by f ::: f p Consider the real variety V 0 := W 0 \ IR n and suppose that (i) V 0 is nonempty and bounded (and hence compact), (ii) the Jacobian J(f ::: f p )(x) is of maximal rank in all points x of V 0 (ie, V 0 is a smooth subvariety of IR n given by the reduced regular sequence f ::: f p ), (iii) the variables X ::: X n are in generic position with respect to f ::: f p Further, let C be any connected component of the compact set V 0, and let b := (a ::: a p; a p ::: a n; a n ) C be a locally maximal point of the last coordinate X n in the non{empty compact set C V 0 Without loss of generality wemay assume that the upper left (p ; ) {minor m of the Jacobian J(f ::: f p ) does not vanish in b (by our assumptions there must be a (p;) {minor of J(f ::: f p ) not vanishing at b ) In any local parametrization of V 0 at b the variable X n cannot be an independent variable, since X n attains a local maximum in b ( a n is this local maximum) Hence, without loss of generality we may assume that the local parametrization of V 0 in b has the following form: there exists an open set U IR n;p containing the point a := (a p ::: a n; ), and a continuously dierentiable function such that ' : U! IR p ':= (' ::: ' p; ' n ) x = ' (x p ::: x n; ) ::: x p; = ' p; (x p ::: x n; ) x n = ' n (x p ::: x n; ) holds for any x =(x p ::: x n; ) U With respect to this local parametrization, the polynomials f k k p induce real valued functions of the form: ~f k (X p ::: X n; ):=

18 POLAR VARIETIES 8 f k (' (X p ::: X n; ) ::: ' p; (X p ::: X n; ) X p ::: X n; ' n (X p ::: X n; )): For every k p and every p j n ; one has the f ~ k k + p; n =0 @X j in the open set U Considering the (p p) {matrix B := n and observing that B is regular in U,we obtain from () that ; det B(x) j =(Adj B)(x) j j j (x) () holds for any x U (here Adj B denotes the adjoint matrix of the matrix B ) As b is a locally maximal point of X n,wehave j (a) =0: holds for every p j n ; Thus, equation () implies B(n j (b)++ B(n p) j (b) =0 (3) for every p j n ; (here we denote for k p by B(n k) the entry of the adjoint matrix Adj B at the cross point of the k {th column and the last row) Taking into account the particular form of the matrix B, the equation system (3) means that

19 3 REAL EQUATION SOLVING 9 det j (b) p; j (b) = 0 (4) holds for every p j n ; Using our notations for the p {minors of the Jacobian J(f ::: f p ), wereinterprete now the equations (4) as M p (b) =:::= M n; (b) =0: Since by assumption m(b) 6= 0 holds, Proposition?? implies that b belongs to the localized determinantal variety ( n;p ) m Therefore, we have b W 0 \ ( n;p ) m, ie, the last formal polar variety W n;p contains the point b On the other hand b is nonsingular point of W 0 and belongs therefore to fw n;p = W n;p n SingW 0 Thus Wn;p f is a non{empty set of dimension zero that contains the real point b of the arbitrarily chosen connected component C of the real variety V 0 In particular, we have forany i n ; p that b Wn;p f \ IR n W i \ IR n = V i holds These considerations imply the following result: Theorem 0 Let W 0 := V (f ::: f p ) be as in Theorem?? If the real variety V 0 := W 0 \IR n is non{empty, bounded and smooth, and if the variables X ::: X n are in generic position with respect to f ::: f p, then every real formal polar variety V i = W i \ IR n i n ; p is a non{empty, smooth manifold of dimension n ; (p + i) and contains at least one representative point of each connected component of the real variety V 0 3 Real Equation Solving The geometric results of Section allow us to design a new ecient procedure that nds at least one representative pointineach connected component of agiven smooth, compact, real complete intersection variety This procedure will be formulated in the algorithmic (complexity) model of (division-free) arithmetic circuits and networks (arithmetic-boolean circuits) over the rational numbers Q Roughly speaking, a division-free arithmetic circuit over Q is an algorithmic device that supports a step by step evaluation of certain (output) polynomials belonging to Q [X ::: X n ], say f ::: f p Each step of

20 3 REAL EQUATION SOLVING 0 corresponds either to an input from X ::: X n,toa constant (circuit parameter) from Q or to an arithmetic operation (addition/subtraction or multiplication) We represent the circuit by a labelled directed acyclic graph (dag) The size of this dag measures the sequential time requirements of the evaluation of the output polynomials f ::: f p performed by the circuit A (division-free) arithmetic network over Q is nothing but an arithmetic circuit that additionally contains decision gates comparing rational values or checking their equality, and selector gates depending on these decision gates Arithmetic circuits and networks represent non{uniform algorithms, and the complexity of executing a single arithmetic operation is always counted at unit cost Nevertheless, by means of well known standard procedures our algorithms will always be transposable to the uniform random bit model and they will be practically implementable as well All this can be done in the spirit of the general asymptotic complexity bounds stated in Theorem?? below Let us also remark that the depth of an arithmetic circuit (or network) measures the parallel time of its evaluation, whereas its size allows an alternative interpretation as "number of processors" In this context we would like to emphasize the particular importance of counting only nonscalar arithmetic operations (ie,only essential multiplications), taking Q { linear operations (in particular, additions/subtractions) for cost{free This leads to the notion of nonscalar size and depth of a given arithmetic circuit or network It can be easily seen that the nonscalar size determines essentially the total size of (which takes into account all operations) and that the nonscalar depth dominates the degree and height of the intermediate results of An arithmetic circuit (or network) becomes a sequential algorithm when we play on it a so{called pebble game By means of pebble games we are able to introduce a natural space measure in our algorithmic model and along with this, a new, more subtle sequential time measure If we play a pebble game on a given arithmetic circuit, we obtain a so{called straight line program (slp) In the same way we obtain from a given arithmetic network a computation tree For more details on our complexity model we refer to [?], [?], [?], [?], [?], [?] and especially to [?] (where also the implementation aspect is treated) In the next Theorem?? we are going to consider families of polynomials f ::: f p belonging to Q [X ::: X n ], for which we arrange the following assumptions and notations: (i) f ::: f p form a regular sequence in Q [X ::: X n ], (ii) for every k p the ideal (f ::: f k ) generated by f ::: f k Q [X ::: X n ] is radical, in

21 3 REAL EQUATION SOLVING (iii) the variables X ::: X n polynomials f ::: f p are in generic position with respect to the Let W 0 := fx C n jf (x) = = f p (x) =0g and denote by SingW 0 the singular locus of W 0 For i n ; p let W i be the i {th formal formal polar variety associated with W 0 and the variables X p+i ::: X n, and let fw i := W i n SingW 0 be the i {th polar variety of W 0 in the usual sense (see Section for precise denitions) We call := maxfmaxfdeg V (f ::: f k ) n SingW 0 j k pg maxfdeg f Wi j i n ; pgg: the degree (of the real interpretation) of the polynomial equation system f ::: f p Finally, let us make the following assumption: (iv) the specialized Jacobian J(f ::: f p )(x) has maximal rank in any point x of V 0 := W 0 \ IR n = fx IR n jf (x) = = f p (x) =0g and V 0 is a bounded semialgebraic set (hence, V 0 is empty or a smooth, compact real manifold of dimension n ; p see Section for details) Theorem Let n p d L and ` be natural numbers with ; d and p n There exists an arithmetic network N over Q of size n p; L(nd) O() and nonscalar depth O(n(log nd + `) log ) with the following property: let f ::: f p be a family of n {variate polynomials of a degree at most d and assume that f ::: f p are given by a division{free arithmetic circuit in Q[X ::: X n ] of size L and nonscalar depth ` Suppose that the polynomials f ::: f p satisfy the conditions (i), (ii), (iii) and (iv) above Further, suppose that the degree of the real interpretation of the polynomial system f ::: f p is bounded by (let us now freely use the notations just before introduced) The algorithm represented by the arithmetic network N starts from the circuit as input and decides rst whether the complex variety Wn;p f is empty If this is not the case, then Wn;p f is a zero{dimensional complex variety and the network N produces an arithmetic circuit in Q of asymptotically the same size and nonscalar depth as N, which represents the coecients of n + univariate polynomials q p ::: p n Q[X n ] satisfying the following conditions: deg q =#f Wn;p maxfdeg p k j k ng < deg q fw n;p = f(p (u) ::: p n (u))ju C q(u) =0g: Moreover, the algorithm represented by the arithmetic network N decides whether the set f Wn;p \IR n is empty If this is not the case, then N produces at most # f Wn;p sign sequences belonging to the set f; 0 g such

22 3 REAL EQUATION SOLVING that these sign sequences encode the real zeroes of the polynomial q "a la Thom" ([?]) In this way N describes the nite, non{empty set Wn;p f \ IR n which contains at least one representative point for each connected component of the real variety V 0 = fx IR n jf (x) == f p (x) =0g Proof We shall freely use the notations of Section Any selection of indices i < <i p n and j k p determines a p { minor M(i ::: i p ) and a (p ; ) { minor m(i ::: i p j k) of the Jacobian J(f ::: f p ) in the following way: M(i ::: i p ) is the determinant ofthe (p p) { submatrix of J(f ::: f p ) with columns i ::: i p, and m(i ::: i p j k) is the determinant of the matrix obtained from the former one deleting the row number j and the column number i k There are p ; n p such possible selections Let us x one of them, say i := ::: i p := p j := p k := p Then, using the notations of Section, we have m(i ::: i p j k) =m M(i ::: i p )=M p Let us abbreviate g := mm p From our assumptions on f ::: f p and Theorem?? and Theorem?? of Section we deduce the following facts: for any i n ; p the polynomials f ::: f p M p ::: M p+i; have degree at most pd They generate the trivial ideal or form a regular sequence in the localized Q -algebra Q[X ::: X n ] g In either case the ideal generated by f ::: f p M p ::: M p+i; in Q [X ::: X n ] g is radical and denes a complex variety that is empty or of degree deg(w i n V (g)) deg(w i n SingW 0 ) = deg f Wi : Moreover, by assumption, the polynomials f ::: f p form a regular sequence in Q [X ::: X n ] g and for each k p the ideal generated by f ::: f k in Q [X ::: X n ] g is radical and denes a complex variety of degree deg(v (f ::: f k ) n V (g)) deg(v (f ::: f k ) n SingW 0 ) : One sees easily that the polynomials f ::: f p M p ::: M n; and g can be evaluated by a division{free arithmetic circuit of size O(L + n 5 ) and nonscalar depth O(log n + `) Applying now for each i n ; p the algorithm underlying [?], Proposition 8 in its rational version [?], Theorem 9 to the system f =0 ::: f p =0 M p =0 ::: M p+i; =0 g 6= 0 we are able to check whether the particular system f =0 ::: f p =0 M p =0 ::: M n; =0 g 6= 0 has a solution in C n If this is the case, then this system denes a zero{ dimensional algebraic set, namely W n;p nv (g), and the algorithm produces

23 3 REAL EQUATION SOLVING 3 an arithmetic circuit in Q which represents the coecients of n + univariate polynomials q p ::: p n Q [X n] satisfying the following conditions: deg q =#(W n;p n V (g)) maxfdeg p k j k ng < deg q W n;p n V (g) =f(p (u) ::: p n(u))ju C q (u) =0g: The algorithm is represented by an arithmetic network of size L(nd) O() and nonscalar depth O(n(log nd+`) log ) and the circuit has asymptotically the same size and nonscalar depth Running this procedure for each selection i < <i p n and j k p we obtain an arithmetic network N 0 of size p ; n ; p L(nd) O() n = p; L(nd) O() and nonscalar depth O(n(log nd + `)log) which decides whether Wn;p f = W n;p n SingW 0 is empty Suppose that this is not the case Then N 0 describes locally the variety f Wn;p which isnow zero-dimensional Each local description of f Wn;p contains an arithmetic circuit representation of the coecients of the minimal polynomial of the variable X n with respect to the corresponding local piece of Wn;p f Moreover, one easily obtains the same type of information for any linear form X i + X n and any variable X i with i < n One multiplies now all minimal polynomials of the variable X n obtained in this way Making this product squarefree (see eg [?], Lemma ) one obtains the polynomial q of the statement of Theorem?? Doing the same thing for the minimal polynomials of each linear form X i + X n and each variable X i with i < n, yields by means of [?], Lemma 6 the polynomials p ::: p n of the statement of Theorem?? All this can be done by means of an arithmetic network N, which extends N 0 and has asympotically the same size and nonscalar depth The nal arithmetic network N is now obtained from N in the same way asintheproof[?], Theorem 8 Remark (i) Using the rened algorithmic techniques of [?] or[?] it is not too dif- cult to see that for inputs f ::: f p represented by straight{line programs of length T and space S the arithmetic network N can be converted into an algebraic computation tree which solves the algorithmic problem of Theorem?? in time O((Tdn + n 5 ) 3 log 3 log log ) and space O(Sdn ) (ii) The smooth, compact hypersuface case (with p := ) of Theorem?? corresponds exactly to [?], Theorem 8 (iii) Let J(f ::: f p ) T denote the transposed matrix of the Jacobian J(f ::: f p ) of the polynomials f ::: f p in the statement of Theorem?? and let D := det J(f ::: f p )J(f ::: f p ) T :

24 3 REAL EQUATION SOLVING 4 From the well{known Cauchy{Binet formula one deduces easily that with the notations of Section the identity D = X i<<i pn det M(i ::: i p ) holds Replacing now in the statement and the proof of Theorem?? for i n ; p the polar variety Wi f by Wi c := W i n V (D) and the parameter by b := maxfmaxfdeg V (f ::: f k ) n SingW 0 j k pg maxfdeg c Wi j i n ; pgg one obtains a somewhat improved complexity result, since b holds Let us now suppose that the polynomials f ::: f p Q[X ::: X n ] satisfy the conditions (i), (ii), (iii), (iv) above Unfortunately the complexity parameter of Theorem?? is strongly related to the complex degrees of the polar varieties W f ::: f Wn;p of W 0 = fx C n jf (x) = = f p (x) = 0g and not to their real degrees Under some additional algorithmic assumptions, which wearegoing to explain below, we may replace the complexity parameter by a smaller one that measures only the real degrees of the polar varieties W f ::: f Wn;p We shall call this new complexity parameter the real degree of the equation system f ::: f p and denote it by Let i n ; p and let us consider the decomposition of the polar varity fw i in irreducible components with respect to the Q {Zariski topology of C n, say f Wi = C [[C s We call an irreducible component C k k s, real if C k \ IR n contains a smooth point of C k The union of all real irreducible components of f Wi is called the real part of f Wi and denoted by Wi We call deg Wi the real degree of the polar variety Wi f Finally, we dene the real degree of the equation system f ::: f p as := maxfmaxfdeg V (f ::: f k ) n SingW 0 j k pg maxfdeg W i j i n ; pgg: We are going to restate the main outcome of Theorem?? in terms of the new complexity parameter For this purpose we have to include in our algorithmic model the following two subroutines: the rst subroutine we need is a factorization algorithm for univariate polynomials over Q In the bit complexity model the problem of factorizing univariate polynomials over Q is known to be of polynomial time complexity [?], whereas in the arithmetic model we are considering here this question is more intricate [?] In the extended

25 3 REAL EQUATION SOLVING 5 complexity model we are going to consider here, the cost of factorizing a univariate polynomial of degree D over Q (given by its coecients), is accounted as D O() the second subroutine allows us to discard non-real irreducible components of the occuring complex polar varieties This second subroutine starts from a straight-line program for a single polynomial in Q[X ::: X n ] as input and decides whether this polynomial has a real zero (however without actually exhibiting it if there is one) Again this subroutine is taken into account at polynomial cost We call an arithmetic network over Q extended if it contains extra nodes corresponding to the rst and second subroutine Modifying our algorithmic model in this way, we are able to formulate the following complexity result which generalizes [?], Theorem and improves the complexity outcome of our previous Theorem?? Remark 3 Let n p d L and ` be natural numbers with d and ; p n There exists an extended arithmetic network N n over Q of size L(nd ) O() p; with the following property: let f ::: f p be a family of n {variate polynomials of a degree at most d and assume that f ::: f p are given by a division{free arithmetic circuit in Q [X ::: X n ] of size L Suppose that the polynomials f ::: f p satisfy the conditions (i), (ii), (iii) and (iv) contained in the formulation of Theorem?? Let us now freely use the notations introduced in the present section Assume that the real variety V 0 = fx IR n jf (x) = = f p (x) = 0g is not empty and that the real degree of the polynomial system f ::: f p is bounded by The algorithm represented by thearithmetic network N starts from the circuit as input and decides rst whether the complex variety Wn;p is empty If this is not the case, then Wn;p is a zero{dimensional complex variety and the network N produces an arithmetic circuit in Q of asymptotically the same size as N,which represents the coecients of n + univariate polynomials q p ::: p n Q [X n ] satisfying the conditions deg q =#W n;p maxfdeg p kj k ng < deg q W n;p = f(p (u) ::: p n(u))ju C q (u) =0g: Eachover Q irreducible component of the complex variety Wn;p contains at least one real point characterized by an irreducible factor of the polynomial q The algorithm represented by the network N returns all these points in a codication "a la Thom" Moreover, the non{empty set Wn;p \ IR n contains at least one representative point foreach connected component of the real variety V 0

26 REFERENCES 6 The proof of this remark is a straight{forward adaptation of the arguments of the proof of [?], Theorem (which treats only the hypersurface case with p := ) to the arguments of Theorem?? above Therefore, we omit this proof References [] Bank, B Giusti, M Heintz, J Mbakop, GM Polar varieties, real equation solving and data structures: The hypersurface case J Complexity 3, No, 5-7, (997), Best Paper Award J Complexity 997 [] Bank, B Giusti, M Heintz, J Mandel, R Mbakop, G M: Polar Varieties and Ecient Real Equation Solving: The Hypersurface Case to appear in: Proceedings of the 3rd Conference Approximation and Optimization in the Caribbean, in: Aportaciones Matematicas, Mexican Society of Mathematics, J Bustamante, M A Jimenez et al (eds) [3] A I Bavinok: Feasibility testing for systems of real quadratic equations, Manuscript, Royal Institute of Technology, Stockholm (99) [4] S Basu, R Pollack, M-F Roy: On the Combinatorial and Algebraic Complexity of Quantier Elimination JACM 43, No 6, ,(996) [5] S Basu, R Pollack, M-F Roy: Complexity of computing semialgebraic descriptions of the connected components of a semialgebraic set Proceedings of ISSAC '98, Gloor, Oliver (ed), Rostock, Germany, August 3{5, 998 New York, NY: ACM Press 5-9 (998) [6] W Baur, V Strassen: The complexity of partial derivatives, Theoret Comput Sci (98) [7] M Ben-Or, D Kozen, J Reif: The complexity of elementary algebra and geometry, J Comput Syst Sci 3 (986) 5-64 [8] M Brodmann: Algebraische Geometrie, Birkhauser Verlag Basel- Boston-Berlin (989) [9] B Buchberger, Ein algorithmisches Kriterium fur die Losbarkeit eines algebraischen Gleichungssystems, Aequationes math 4 (970) 37{383 [0] P Burgisser, M Clausen, M A Shokrollahi Algebraic complexity theory With the collaboration of Thomas Lickteig Grundlehren der Mathematischen Wissenschaften 35 Berlin: Springer xxiii, 68 (997)