University of California. November 16, Abstract

Transcription

1 On the Complexity of Sparse Elimination æ Ioannis Z Emiris Computer Science Division University of California Berkeley, CA 90, USA emiris@csberkeleyedu November 1, 199 Abstract Sparse elimination exploits the structure of a set of multivariate polynomials by measuring complexity in terms of Newton polytopes We examine polynomial systems that generate 0-dimensional ideals: a generic monomial basis for the coordinate ring of such a system is deæned from a mixed subdivision We oæer a simple proof of this known fact and relate the computation of a monomial basis to the calculation of Mixed Volume The proof relies on the construction of sparse resultant matrices and leads to the eæcient computation of multiplication maps in the coordinate ring and the calculation of common zeros It is shown that the size of monomial bases and multiplication maps in the context of sparse elimination theory is a function of the Mixed Volume of the Newton polytopes, whereas classical elimination considers simply total degree Our algorithm for the sparse resultant and for root-ænding has worst-case complexity proportional to the volume of the Minkowski Sum of these polytopes We derive new bounds on the Minkowski Sum volume as a function of the Mixed Volume and use these results in order to give general upper bounds on the complexity of computing monomial bases, sparse resultants and common zeros 1 Introduction Sparse elimination theory generalizes several results of classical elimination theory on multivariate polynomial systems by considering the structure of the given polynomials, namely their coeæcients which are a priori zero and their Newton polytopes This leads to stronger algebraic and combinatorial results in general, whose complexity depends on eæective rather than total degree The foundations were laid in the work of Gelfand, Kapranov and Zelevinsky ë1, 1ë The central object in elimination theory is the resultant, which characterizes the solvability of an overconstrained system A generalization of the Sylvester resultant for two univariate polynomials is the sparse resultant for an arbitrary number of multivariate polynomials, which, in many cases, has lower degree than its classical counterpart, since its degree depends on the Bernstein bound ëë as explained in the next section Bernstein's bound is at most equal to Bezout's bound on the number of roots for an n æ n polynomial system and for sparse systems it is often smaller; the comparison between the two approaches is formalized in the following section Eæective algorithms for the construction of compact matrix formulae for the sparse resultant already exist We rely on the construction of ëë in order to oæer a simple proof of the fact that a mixed subdivision deænes a monomial basis for the coordinate ring of the given polynomial system We consider the important case of square polynomial systems, ie systems of n polynomials in n variables One approach to the solution of such systems is based on the construction of multiplication maps in the respective coordinate ring and the latter problem requires the computation of monomial bases This paper proves upper bounds on the worst-case asymptotic bit complexity of these three problems, starting with monomial bases, æ Part of the results appeared in preliminary form in: IZ Emiris and A Rege, Monomial Bases and Polynomial System Solving, in Proc ACM Intern Symp on Symbolic and Algebraic Computation, 199, pp 11í1 Supported by a David and Lucile Packard Foundation Fellowship and by NSF PYI Grant IRI-898

2 y x Figure 1: The Newton polytope of polynomial c 1 y + c x y + c x y + c x + c xy The dotted triangle is the Newton polytope of the dense polynomial of the same total degree continuing with the implications on multiplication maps and concluding with root-ænding Throughout, we emphasize the relevance of Mixed Volume as a measure of the inherent complexity, while the complexity of our algorithms is mostly dependent upon the volume of the Minkowski Sum A central issue in the analysis, thus, becomes the relation of Mixed Volume to Minkowski Sum, which we tackle in a general setting before establishing the worst-case asymptotic complexity bounds Generically, a square polynomial system has a ænite number of isolated and distinct roots, so we restrict attention to this case when considering monomial bases Namely, the given polynomials deæne a radical ideal whose variety is 0-dimensional For system solving only the latter hypothesis is required since there exist techniques for coping with non-radical ideals Sparse resultants have a signiæcant potential for applications reducing to questions in elimination and to polynomial system solving Techniques based on ad-hoc resultants have led to impressive results on certain problems in inverse kinematics, graphics and modeling ë, ë Currently, problems from computer vision, direct kinematics and molecular structure are being successfully solved by the general sparse elimination methods discussed in this paper, thus illustrating their practical relevance ë1, 8ë We start with an introduction to the theory of sparse elimination in the next section and we continue with a comparative exposition of previous work in Section and a more detailed presentation of an eæcient resultant matrix construction in Section The deænition of monomial bases through mixed subdivisions is presented in Section, then a more eæcient way of deæning them is shown equivalent to the original one and an algorithm for their computation is presented Section proves how monomial bases specify multiplication maps and Section shows how the latter allow polynomial system solving by two alternative ways We relate Minkowski Sum volumes to Mixed Volumes in Section 8 and use these results in Section 9 to formalize general upper bounds on the complexity of constructing monomial bases and sparse resultant matrices as well as of solving polynomial systems Section 10 concludes with some open questions Sparse Elimination Theory Sparse elimination theory considers Laurent polynomials in n variables, where the exponents are allowed to be arbitrary integers The polynomial ring is Këx 1 ;x,1 ;:::;x 1 n;x,1 n ë=këx; x,1 ë, for some base æeld K We shall be interested in polynomial roots in èk æ è n, where K is the algebraic closure of K and K æ = K nf0g Deænition 1 Let f be apolynomial in Këx; x,1 ë The ænite set Aç Z n of all monomial exponents corresponding to nonzero coeæcients is the support of f The Newton polytope of f is the convex hull of A, denoted Q = ConvèAè ç IR n If we use x e to denote the monomial x e1 1 :::xen n, where e =èe 1;:::;e n è Z n X is an exponent vector, then f = c j x aj ; 8c j =0: a ja Newton polytopes model the sparse structure that we wish to exploit in polynomials Fig 1 depicts the Newton polytope for a bivariate polynomial and compares it with the Newton polytope of the dense polynomial with the same total degree, ie a polynomial in which every coeæcient is nonzero Newton polytopes provide a bridge from algebra to geometry since they permit certain algebraic problems to be cast in geometric terms Thus we need some concepts from polytope theory

3 Deænition The Minkowski Sum A + B of convex polytopes A and B in IR n is the set A + B = fa + b j a A; b Bg ç IR n : It is easy to prove that A + B is a convex polytope ëë Deænition Given convex polytopes A 1 ;:::;A n ç IR n, there is a unique, up to multiplication by a scalar, real-valued function MV èa 1 ;:::;A n è,called the Mixed Volume of the given polytopes, which is multilinear with respect to Minkowski addition and scalar multiplication, ie for ç; ç IR ç0 and convex polytope A 0 k ç IRn MV èa 1 ;:::;ça k + ça 0 k;:::;a n è=çmv èa 1 ;:::;A k ;:::;A n è+çmv èa 1 ;:::;A 0 k;:::;a n è: To deæne Mixed Volume exactly we require that MV èa 1 ;:::;A 1 è=n!v èa 1 è; where V èæè is the standard n-dimensional volume function An equivalent deænition ëë is Deænition For ç 1 ;:::;ç n IR ç0 and convex polytopes A 1 ;:::;A n ç IR n, the Mixed Volume MV èa 1 ;:::;A n è is precisely the coeæcient of ç 1 ç æææç n in V èç 1 A 1 + æææ+ ç n A n è expanded asapolynomial in ç 1 ;:::;ç n We now study systems of n Laurent polynomials inn variables Let f 1 ;:::;f n Këx; x,1 ë be the polynomials and A i, Q i the support and Newton polytope of f i A system is called unmixed when all supports are identical; otherwise it is mixed This article is concerned with the latter and more general case The shorthands MV èf 1 ;:::;f n è and MV èa 1 ;:::;A n è are occasionally used for the Mixed Volume MV èq 1 ;:::;Q n è The Newton polytopes oæer a convenient model for the sparseness of a polynomial system, in light of Bernstein's upper bound on the number of common roots This bound is also called the BKK bound to underline the contributions of Kushnirenko and Khovanskii in its development and proof ë1, 19ë Theorem ëë Let f 1 ;:::;f n Këx 1 ;x,1 ;:::; x n ;x,1 n ë with Newton polytopes Q 1 ;:::;Q n The number of isolated common zeros in èk æ è n, multiplicities counted, is either inænite, or does not exceed MV èq 1 ;:::;Q n è For almost all specializations of the coeæcients the number of common zeros is exactly MV èq 1 ;:::;Q n è Interesting extensions to this theorem concern the weakening of the genericity condition ëë and the case of roots in èkè n ë, ë We state the latter result Theorem ëë For polynomials f 1 ;:::;f n Cëx; x,1 ë with supports A 1 ;:::;A n the number of common isolated zeros in C n,counting multiplicities, is upwards bounded bymv èa 1 ëf0g;:::;a n ëf0gè The Mixed Volume is typically signiæcantly lower than Bezout's bound, which bounds the number of projective solutions by Q i deg f i, where deg f i is the total degree of f i One example is the simple and generalized eigenproblems on n æ n matrices The Bezout bound in both cases is n+1, while the exact number of right eigenvector and eigenvalue pairs is n, which is exactly given by the Mixed Volume The two bounds coincide for dense polynomials, because each Newton polytope is an n-dimensional unit simplex scaled by deg f i By deænition, the Mixed Volume of the dense system is MV èdeg f 1 S;:::;deg f n Sè= Y i deg f i MV ès;:::;sè= Y i deg f i ; where S is the unit simplex in IR n with vertex set fè0;:::;0è; è1; 0;:::;0è;:::;è0;:::;0; 1èg P n+1 A technical assumption is that, without loss of generality, the aæne lattice generated by A i=1 i is n- dimensional This lattice is identiæed with Z n possibly after a change of variables, which can be implemented by computing the appropriate Smith's Normal form The central object in elimination is the resultant ofn + 1 polynomials inn variables It is a single polynomial in the polynomial coeæcients which characterizes the existence of nontrivial common zeros In sparse elimination, nontrivial roots lie in èk æ è n and the sparse resultant ofanoverconstrained system is deæned as follows ë0ë Let c be the vector of all polynomial coeæcients, regarded as indeterminates, and let Z 0 be the set of all such vectors c for which the polynomials have a common zero Let Z be the Zariski closure of Z 0

4 Deænition ëë The sparse resultant R = RèA 0 ;:::;A n è of polynomials f 0 ;f 1 ;:::;f n Këx; x,1 ë is an irreducible polynomial in Zëcë IfcodimèZè =1then R is the deæning polynomial of hypersurface Z IfcodimèZè é 1 then R =1Furthermore, the degree ofr in the coeæcients of polynomial f i equals MV èf 0 ;:::;f i,1 ;f i+1 ;f n è, for i =0;:::;n Some authors call this the Newton resultant to underline its dependence on the Newton polytopes interesting to note that it subsumes the classical deænition of the resultant ëë It is Related Work A method for constructing generic vector bases of coordinate rings as monomials indexed by the lattice points in the mixed cells of a mixed subdivision was ærst demonstrated by Pedersen and Sturmfels ë1ë The term mixed monomial bases highlights the fact that they apply to arbitrary systems and that they are obtained through a mixed subdivision A crucial hypothesis is that the given polynomials are generic, which is also assumed here Our approach is based on a matrix formula for the sparse resultant ëë which leads to an immediate proof and applies also to arbitrary systems Under appropriate choice of the various parameters our approach obtains the same bases Sparse resultants have been studied by several authors and eæective methods for the construction of matrix formulae have been proposed in ë,, 11, ë The ærst eæcient and general method ëë is sketched in the next section The heuristic in ë11ë takes a diæerent tack in an eæort to improve upon the upper bounds, namely by avoiding the extraneous factor; it has been implemented and has given some encouraging preliminary results ë1ë Exact matrix formulae for particular classes of polynomial systems are suggested in ëë; they are called of Sylvester-type since they generalize the Sylvester determinant for two univariate polynomials Root-ænding methods based on matrices have a long history The classical resultant provides a means for root-ænding by the use of U-resultants ë,,, ë The reduction to an eigenvalue and eigenvector problem was formalized in ëë and, independently, in ë, ë The latter articles discuss alternative strategies for dealing with ill-conditioned or singular matrices, some leading to the generalized eigenproblem; this issue is revisited at the end of Section The deænition of monomial bases and multiplication maps is also possible through Gríobner bases, so we can again reduce polynomial system solving to an eigenproblem; this approach is surveyed in ëë The problem of monomial bases is equivalent to computing Mixed Volumes, for which various algorithms have been proposed We relate our proof on monomial bases to the most eæcient general Mixed Volume algorithm to date, originating from Sturmfels' Lifting Algorithm ëë and modiæed by the heuristic proposed by Emiris and Canny ë1ë Empirical results of this algorithm are reported in ë1ë Other methods, exploiting special cases, were proposed in ë1, 9,8ë in conjunction to deæning sparse homotopies for solving polynomial systems by continuation Sparse Resultant Matrices The main construction in our approach for establishing the result on monomial bases and for obtaining the sparse resultant is the construction of a matrix M in the polynomial coeæcients, whose determinant is a nontrivial multiple of the sparse resultant The ærst eæcient algorithm was proposed by Canny and Emiris ëë and subsequently generalized by Sturmfels ëë Given are polynomials f 0 ;:::;f n Këx; x,1 ë Let Q 0 denote the Minkowski Sum of all input Newton polytopes Q 0 = Q 0 + Q 1 + æææ+ Q n ç IR n : We shall deæne a subset of the lattice points in Q 0 that index the rows and columns of M To this end, we adopt a technique from ëë Select n + 1 linear lifting forms l i :IR n! IR for 0 ç i ç n Then deæne the lifted Newton polytopes bq i = fèpi ;l i èp i èè : p i Q i gçir n+1 ; 0 ç i ç n and take their Minkowski sum bq 0 = b Q0 + æææ+ b Qn ç IR n+1 :

5 Given any polytope in IR n+1, its lower envelope with respect to vector è0;:::;0; 1è IR n+1 is the union of all n-dimensional faces, or facets, whose inner normal vector has positive last component In the rest of this article we always consider lower envelopes with respect to vector è0;:::;0; 1è The projection of all facets on the lower envelope of b Q0 onto Q 0 induces a mixed subdivision æ 0 of the latter The linear lifting functions l i are chosen to be suæciently generic, such that every point in the mixed subdivision is uniquely expressed as a sum p = p 0 + p 1 + æææ+ p n : p i Q i : This sum is called an optimal sum because the p i are speciæed by the requirement that their lifted images add up to a point bp on the lower envelope of Q0 b P In other words, they minimize the aggregate lifting function i l ièp i è over all èn + 1è-tuples of points whose sum equals p The genericity requirement for l i is achieved by picking, for i =0;:::;n, a random integer vector of the coeæcients of l i Each entry is independent and uniformly distributed with bit size L l, for some constant L l é 1 Then the probability that the genericity condition fails is bounded by ny Probëfailureë ç i=0 r i =èn Ll è : r i is the vertex cardinality ofq i : è1è For most problems in practice it suæces to use one-word values for the l i coeæcients It is straightforward to check deterministically whether a particular choice of lifting forms satisæes the genericity requirement A consequence of the uniqueness condition on optimal sums for points is that each maximal cell ç in æ 0 is uniquely expressed as a Minkowski sum ç = F 0 + æææ+ F n ç IR n : F i is a face of Q i ;i=0;:::;n: This is called the optimal sum for ç under the speciæc subdivision Maximal cell ç is the projection along è0;:::;0; 1è of a facet on the lower envelope of b Q0 that is uniquely expressed as the Minkowski Sum of those faces in b Qi corresponding to F i A property of mixed subdivisions is that cells are either mixed or unmixed, mixed cells being Minkowski sums such that exactly one face in their optimal sum is a vertex and all others are edges Deænition 1 A mixed maximal cell of the induced mixed subdivision of Q 0 is i-mixed if, in its expression as an optimal sum of faces, the summand from Q i is some vertex a ij : ç = E 0 + æææ+ E i,1 + a ij + E i+1 + æææ+ E n ; where E k is an edge of Q k : It can be shown that, if V èæè denotes n-dimensional volume, X MV èq 0 ;:::;Q i,1 ;Q i+1 ;:::;Q n è= V èçè: i-mixed ç The rows and columns of M are indexed by the integer lattice points E =èq 0 + æè ë Z n ; where Q 0 + æ is a polytope obtained by perturbing Q 0 by some arbitrarily small æ Q n,chosen to be suæciently generic so that every perturbed lattice point lies strictly inside a maximal cell The mixed decomposition, corresponding to æ æ,onq 0 + æ is denoted æ 0 æ The obvious bijection e! xe ;e Z between the integer lattice and the set of Laurent monomials allows us to consider E either as a point set or a monomial set For every p ç, for some cell ç, deæne a row content function RCèæè such that RCèpè =èi; jè if and only if a ij isavertex in the optimal sum of ç and i is the maximum index for which the summand is a vertex Than the row ofm corresponding to p contains the coeæcients of x p,aij f i Coeæcient c ik appears in the column indexed by column monomial x q if c ik x q is a term of x p,aij f i The entries of this row that do not correspond to any column monomial are zero Lemma ëë The above construction of M produces a well-deæned and square matrix with size jej, where jæj denotes set cardinality

6 Wenowsketch the proof establishing the generic nonsingularityofm, ie nonsingularity when the polynomials have generic, or indeterminate, coeæcients Let matrix c M be obtained from M by specializing all coeæcients to powers of a new variable t and denote by c Mpq the entry of c M with row index p and column index q, for some p; q E, then Lemma ë, Lemma 1ë For all non-zero elements c Mpq with p = q, deg t èc Mpq è é deg t èc Mqq è Lemma Every principal minor of M is generically nonzero Proof Let N be the square submatrix of M corresponding to a given principal minor and let b N be the corresponding submatrix of MIf c Npq b is the entry indexed by p; q E, then det N b Y = q bn qq + higher order terms in t; the product being over all q E indexing the rows of N: By the previous lemma, this term does not vanish for suæciently small positive t, hence det b N is nonzero Now det N equals the product of det b N multiplied by apower in t, therefore it is also generically nonzero This also implies that M is generically nonsingular We can now formalize the properties of M Theorem ëë Matrix M is well-deæned, square, generically nonsingular and its determinant is divisible by the sparse resultant Rèf 0 ;:::;f n è Moreover, the degree ofdet M in the coeæcients of f 0 equals MV èf 1 ; :::;f n è, while its degree in the coeæcients of f i for i =1;:::;n is greater or equal to MV èf 0 ;:::;f i,1 ;f i+1 ;f n è From Deænition the degree of det M is exact in f 0 whereas an extraneous factor in the coeæcients of f 1 ;:::;f n may exist For ænding all isolated roots of polynomial systems an exact expression for the sparse resultant is not required so we use det M to compute a superset of the roots M generalizes the classical Macaulay matrix since it reduces to the latter on dense systems A greedy variant of this algorithm that typically leads to smaller matrices has been implemented by J Canny and P Pedersen and described in ë1ë The construction of M leads to the explicit construction of the sparse resultant R by two alternative methods discussed in ë, 8ë Monomial Bases for Coordinate Rings For n generic Laurent polynomials f 1 ;:::;f n in n variables, the deænition of monomial bases from mixed subdivisions was ærst demonstrated by Pedersen and Sturmfels ë1ë Their proof relies on reducing the general problem to binomial systems via Puiseux series Theorem veriæes their result However, we use a diæerent proof which is considerably simpler once the construction of resultant matrix M is established and which leads, in the next section, to a constructive approach for ænding the common zeros The genericity of the polynomials is equivalent to saying that all coeæcients are generic so we regard them as indeterminates Let I = Ièf 1 ;:::;f n è be the ideal that they generate and V = V èf 1 ;:::;f n è èk æ è n their variety, where K is the algebraic closure of æeld K Assume that V has dimension zero Then its coordinate ring Këx; x,1 ë=i is an m-dimensional vector space over K by Theorem, where m = MV èf 1 ;:::;f n è=mv èq 1 ;:::;Q n è: In addition, the ideal I = Ièf 1 ;:::;f n è is assumed to be radical, or self-radical, ie I = p I, which is equivalent to saying that all roots in V are distinct We add a generic f 0 Këx; x,1 ë to the set f 1 ;:::;f n and deæne the Minkowski sum Q 0 + æ and its mixed subdivision æ 0 æ as in the previous section Without loss of generality we can choose f 0 such that it has the constant monomial 1 as one of its monomials This follows easily from the fact that given an arbitrary f 0 in Këx; x,1 ë, we can divide it by one of its monomials without changing its roots in èk æ è n Let BçEç Z n be the set of all integer lattice points that lie in 0-mixed cells, in æ 0 æ Equivalently, B is the set of all Laurent monomials with exponent vectors in the 0-mixed cells By Theorem, jbj = m and we can write B = fb 1 ;:::;b m g

7 An important property of the matrix construction of the previous section is that postmultiplication with certain column vectors expresses evaluation of the polynomials whose coeæcients have ælled in the rows of the matrix More precisely, for an arbitrary æ K n, M æ q = æ p f ip èæè ; èè where p Eindexes the row ofm that contains the coeæcients of x p f ip èxè and q Eindexes the column corresponding to monomial x q Since A 0 contains 0 n Z n we can always pick, without loss of generality, lifting function l 0 such that Q 0 contributes only its zero vertex 0 n as a summand to the 0-mixed cells The proof of Lemma formalizes the requirement onl 0 and proves the feasibility of this construction By deænition, every row indexed by a monomial in B contains the coeæcients of x b,0n f 0 = x b f 0, for some b B The partition of E into B and EnBdeænes four blocks in M shown below, where the rightmost set of columns and bottom set of rows are indexed by B Submatrices M 11 and M are square of size je n Bj = jej, m and jbj = m respectively, while M 1 and M 1 are rectangular Let æ V be a æxed common root Relation èè becomes M æ qc æ bi = M 11 M 1 M 1 M æ qc æ bi = 0 æ bi f 0 èæè where q c ranges over EnBand b i ranges over B By Lemma every principal minor of M is generically nonzero, hence the inverse submatrix M,1 exists 11 Then, we can deæne m æ m matrix M 0 = M, M 1 M,1 M 11 1: èè Lemma 1 Assume that variety V = V èiè has dimension zero, ideal I is radical and B = fb 1 ;:::;b m g is the set of points in 0-mixed cells in æ 0 æ Then, all eigenvectors of M 0 are of the form ëæ b1 ;:::;æ bm ë for some root æ V Proof We premultiply both sides of èè with the non-singular matrix ç I 0,M 1 M,1 11 I where I stands for the identity matrix of appropriate size, and obtain M 11 M 1 0 M 0 The bottom part of this matrix equation is of interest: M 0 æ b1 æ bm æ qc æ bi = = æ b1 f 0 èæè æ bm f 0 èæè èè ç ; èè 0 æ bi f 0 èæè : : èè

8 Let v 0 æ be the column vector ëæ b1 ;:::;æ bm ë, with b i B Since æ èk æ è n,every v 0 æ belongs to èk æ è m, namely, it is nonzero; furthermore, èè yields an eigenvector equation M 0 v 0 æ = f 0 èæèv 0 æ è èm 0, f 0 èæèiè v 0 æ =0: Since there are exactly m roots and we can construct one such vector per root, we obtain m such vectors This is the largest possible number of eigenvectors, hence all eigenvectors of M 0 are of this form Theorem Assume that variety V = V èiè has dimension zero, ideal I is radical and B is the set of monomials corresponding to integer lattice points in 0-mixed cells in the subdivision æ 0 æ of Q0 Then B forms a vector-space basis for the coordinate ring Këx; x,1 ë=i over K P n Proof Let f 0 èxè =c 00 + c j=1 0jx j Këx; x,1 ë with c 00 ;:::;c 0n being generic indeterminates The roots æ are distinct and, by the genericity ofc 0j, all eigenvalues f 0 èæè are distinct This implies that all eigenvectors væ 0 are linearly independent If the monomialsinb are not a basis of Këx; x,1 ë=i, then a non-trivial linear combination of them P over K must m belong to I Hence, there are elements k 1 ;:::;k m K not all zero such that, for every æ V, Construct now the square matrix below with v 0 æ j this matrix has dependent rows: =ëæ b1 j ;:::;æbm æ b1 1 æ b1 æææ æ b1 m æ b 1 æ b æææ æ b m èè k i=1 iæ bi =0 j ë as the j-th column, where V = æ 1 ;:::;æ m ; : è8è æ bm 1 æ bm æææ æ bm m This contradicts the independence of vectors v 0 æ j so B is indeed a basis In other words, we have deæned a canonical surjective homomorphism Këx; x,1 ë! Këx; x,1 ë=i : g! g mod I = X b ib c bi x bi ;c bi K such that g I, c bi =0; 8 b i B: In words, every polynomial g is mapped to the canonical representative of its coset with respect to ideal I It turns out that we can compute the basis without going through the resultant matrix because the set B is deæned independently of f 0 Consider a mixed subdivision æ æ of the perturbed Minkowski sum Q + æ = Q 1 + æææ+ Q n + æ induced by l 1 ;:::;l n, where both l i and æ Q n are the same as above The subdivision is speciæed by deæning Minkowski Sum bq = b Q1 + æææ+ b Qn ç IR n+1 of the lifted Newton polytopes b Qi and projecting its lower envelope facets onto the maximal cells of æ æ The maximal cells in the subdivision are again either mixed, when they are the Minkowski sum of n edges, or unmixed The sum of all mixed cell volumes is m = MV èf 1 ;:::;f n è Lemma Consider the mixed subdivision æ æ of Q + æ induced by lifting forms l 1 ;:::;l n Then the set B of points in the 0-mixed cells of æ 0 æ equals the set of all integer lattice points in the mixed cells of æ æ Proof Recall that Q 0 is the Minkowski sum of n + 1 Newton polytopes, A 0 contains the zero exponent 0 n and æ 0 æ is the mixed decomposition of Q0 + æ induced by l 0 ;l 1 ;:::;l n Any point on the lower envelope of b Q0 is of the form bp + ba 0j, where bp is on the lower envelope of b Q and ba 0j b Q0 We wish to show that every such point, for appropriate l 0, has a unique summand from b Q0, namely the lifted image of 0 n Consider points bp; bq on the lower envelope of Q b and assume that bp +è0 n ;l 0 è0 n èè and bq + ba 0j lie on the same vertical, for some a 0j =0 n We can pick l 0 suæciently large so that bp +è0 n ;l 0 è0 n èè is on the lower envelope whereas bp + ba 0j is not For this it suæces to require that l 0 èa 0j è é nx i=1 l i èa iji è; 8 a 0j Q 0 ;a 0j =0 n ; 8a iji Q i : è9è 8

9 Consider a lower envelope facet bç of b Q, where its perturbed projection ç + æ is a mixed cell in ææ A similar argument shows that under è9è, for every facet bç, the sum è0 n ;l 0 è0 n èè + bç is a lower envelope facet on b Q0 Then the total volume of all cells in æ 0 æ of the form 0n + ç + æ, where ç + æ is a mixed cell of æ æ,ism All of these cells are 0-mixed by construction, hence there are no more 0-mixed cells in æ 0 æ An appropriate choice of l 0, therefore, establishes a bijective correspondence between mixed cells of æ æ and 0-mixed cells of æ 0 æ The proof is completed by noting that the integer points in the latter cells are of the form 0 n + p, where p Q and, actually, p belongs to a mixed cell of æ æ This immediately leads to an equivalent statement of Theorem Theorem Assume that variety V = V èiè has dimension zero, ideal I is radical and let B be the set of monomials corresponding to integer lattice points in mixed cells in the subdivision æ æ of Q Then B forms a vector-space basis for the coordinate ring Këx; x,1 ë=i over K This gives rise to the following direct algorithm for computing the monomial basis: First, compute the Newton polytopes Q 1 ;:::;Q n Second, pick suæciently generic lifting functions l 1 ;:::;l n and compute the induced mixed subdivision æ æ of Q + æ Third, identify all mixed maximal cells ç of æ æ and, fourth, enumerate all lattice points ç ë Z n for each ç Each of these lattice points is the exponent of a unique monomial in the basis The third step is the main part of the algorithm and, together with the equivalent problem of Mixed Volume computation, has been addressed by several authors as described in Section The main idea of the algorithm from ë, 1ë is to test all edge combinations, each combination including exactly one edge from each Newton polytope: The combinations that pass all tests deæne a mixed cell To prune the search we eliminate edge combinations by inexpensive tests on subsets of these combinations, relying on the observation that an edge combination e 1 ;:::;e k corresponds to a facet on the lower envelope of the respective k lifted polytopes only if the same holds for every subset of these edges Multiplication Maps This section shows how matrix M 0, deæned in èè, is the matrix of the endomorphism in Këx; x,1 ë=i which expresses multiplication by polynomial f 0, hence it provides a multiplication map in Këx; x,1 ë=i Multiplication maps are the essential object in solving polynomial systems by matrix techniques Again, we are assuming that I is radical, the corresponding variety V zero-dimensional, m denotes the cardinality ofv and Këx; x,1 ë=i is an m-dimensional vector space over K Lemma 1 The rows of M 0 contain the coeæcients of polynomials x bi f 0 mod I, for some b i B Proof Premultiplication of M by matrix èè has the eæect of adding scalar multiples of the rows indexed by EnBto those indexed by B Hence, the row ofm X indexed by b i Bnow contains the coeæcients of g = x bi f 0 + k p x p f jp ; for some k p K: penb On the other hand, èè shows that each rowofm 0 corresponds to a polynomial h which is a linear combination of the monomials inb, over K Thus g, h Ior g ç h èmod Iè and the the lemma is proven Since B provides a vector space basis for Këx; x,1 ë=i over K, every polynomial g Këx; x,1 ë=i can be expressed as a row vector v g K m, whose entries are indexed by B and contain the respective coeæcients Theorem Let M 0 denote both the matrix and the associated endomorphism in Këx; x,1 ë=i with respect to basis B Then this endomorphism expresses multiplication by polynomial f 0 Këx; x,1 ë=i, M 0 : Këx; x,1 ë=i!këx; x,1 ë=i : g! gf 0 mod I: In other words, if row vector v g expresses polynomial g Këx; x,1 ë=i, with respect to basis B, then row vector v g M 0 expresses polynomial gf 0 Këx; x,1 ë=i with respect to the same basis 9

10 Proof P From the previous lemma row b i of M 0 contains the coeæcients of polynomial x bi f 0 mod I Let g = m c i=1 ix bi, then gf 0 mod I = = mx i=1 mx i=1 c i èx bi f 0 mod Iè 0 X c m j=1 1 è X m mx! M 0 A ij xbj = x bj c i M 0 ij : j=1 i=1 If b j Bindexes the P j-th column of M 0, then the last polynomial can be expressed as the row vector indexed by m B with j-th entry c i=1 im 0 ij By the deænition of v g we have v g M 0 =ëc 1 ;:::;c m ëm 0 = " mx i=1 c i M 0 i1;:::; mx i=1 c i M 0 im and the claim is established Polynomial System Solving Matrix M 0 essentially allows computation within the coordinate ring This is the essential property in ænding all roots of the given system of polynomials by matrix-based techniques Notice that, although the computation of monomial bases did not require the use of f 0, here we do need this extra polynomial In computing matrix M by the algorithm in ëë, f 0 is linear with generic coeæcients, as in the proof of Theorem In practice, we let one coeæcient be an indeterminate u and we pick random coeæcients c 0j, for j =1;:::;n, from some range of possible integer values of size Ré1, so è ; f 0 = u + c 01 x 1 + æææ+ c 0n x n Këx; x,1 ;uë: This is essentially the U-resultant construction, extensively studied in the context of classical elimination Recall that the resultant characterizes the solvability of the system, therefore the addition of an additional, artiæcial constraint f 0 may P eliminate some of the solutions of f 1 = æææ= f n = 0 unless f 0 includes free variable u, which takes the value, j c 0jæ ij at root æ i =èæ i1 ;:::;æ in è A bad choice for c 01 ;:::;c 0n is one that will result in the same value of f 0, u at two distinct roots æ 1 and æ Assume that æ 1 and æ diæer in their i-th coordinate for some, ié0, then æx all choices of c 0j for j = i; the m probability of a bad choice for c 0i is 1=R, and since there are æ pairs of roots, the total probability of failure for this scheme is ç ç m Probëfailureë ç =r : c 0j f1;:::;rg; j=1;:::;n: It suæces, therefore, to pick c 0j from a suæciently large range in order to make the probability of success arbitrarily high Moreover, it is clear that any choice of f 0 coeæcients can be tested deterministically at the end of the algorithm The construction of M is not aæected by this deænition of f 0 By abuse of notation we write the new multiplication map matrix as M 0 + ui, where M 0 isanumeric matrix, u is the new variable and I is the m æ m identity matrix M 0 is deæned in the same way as before, since no assumptions were made about the coeæcients of f 0 besides their genericity For solving the polynomial system we have to specialize f 0 and separate the matrix entries dependent onu from the numeric matrix Now to deæne an eigenproblem èè becomes, for the i-th root æ i V, ëm 0 +èu, f 0 èæ i èèiëv 0 æ =0è M 0, X j c 0j æ ij 1 A I v 0 æ =0; which implies that the i-th eigenvalue of M 0 is P j c 0jæ ij and the respective eigenvector is the same as before 10

11 If the generated ideal I is radical then every eigenvalue has algebraic multiplicity one We can relax the condition on I by simply requiring that each eigenvalue has geometric multiplicity one Algebraic multiplicity captures the usual notion of multiplicity, whereas geometric multiplicity expresses the dimension of the eigenspace associated with an eigenvalue If there exist eigenvalues of higher geometric multiplicity we can use the properties of the U-resultant to recover the root coordinates ë,,, ë By Lemma 1 each eigenvector væ 0 of M 0 contains the values of monomials B at some common root æ èk æ è n Deæne vector v æ =,M,1 M 11 1v 0 æ è10è of size jej, m, indexed by EnB By construction we obtain the following Lemma 1 The concatenation of vectors v æ and v 0 æ lies in the kernel of the homomorphism deæned by the top jej, m rows of M in èè: M11 M 1 ç v æ v 0 æ ç ç 0 = 0 ç ; è11è where 0 here isazerovector of length jej, m Therefore the element of v æ indexed byp EnBis the value of monomial x p at root æ It follows that vectors v æ and v 0 æ together contain the values of every monomial in E at some root æ Lemma Let p 0 ;p 1 ;:::;p s E, s ç n, be a set of points such that, the matrix with i-th row p i, p 0 has rank n Then, given v æ and v 0 æ,wecan compute the coordinates of root æ V èiè Ifp 0 ;p 1 ;:::;p s Bthen v 0 æ suæces Proof Let P be the s æ n matrix whose i-th row isp i, p 0 By linear algebra, there exists nonsingular s æ s matrix Q such that QP is an upper-triangular matrix with nonzero diagonal d 1 ;:::;d n Z Now consider the column subvector of ëv æ ;væë 0 indexed by points p i, which is in bijective correspondence with the rows of P Apply the sequence of elementary row operations speciæed by Q to the elements of this column subvector as follows: a row swap is an exchange of vector entries, the scaling of a row by c corresponds to raising the respective entry to c and the addition of row i, multiplied by c, torow j corresponds to multiplication of vector entry j by the i-th entry raised to c Let q denote this vector transformation The resulting vector has the last s, n entries equal to 1 Let æ =èæ 1 ;:::;æ n è and e ;e n ;g n Z, then this transformation can be written as follows: QP = d 1 e æææ æææ e n 0 æææ 0 d n,1 g n 0 æææ 0 0 d n 0 æææ 0 0 æææ 0 then q : æ p1,p0 æ ps,p0! æ d1 1 æe ææææen n æ dn,1 n,1 ægn n æ dn n 1 The ænal step consists in reading oæ the coordinates of æ from the modiæed vector For ease of notation assume that no row exchanges were necessary The value of coordinate n is obtained by taking the d n -th root of the n-th entry of the vector The èn, 1è-st entry equals æ dn,1 n,1 ægn n so æ n,1 is the d n,1 -th root of the vector's èn, 1è-st entry divided by æ gn n The rest of the root coordinates are computed in an analogous fashion; this is in a sense the backwards substitution phase where the row elementary operations are transformed so that they apply to the exponents n + 1 points are necessary and suæcient, if aænely independent, to recover all root coordinates E always includes n + 1 such points because the lattice spanned by it has dimension n If the dimension were lower every Newton polytope would have zero volume and all Mixed Volumes would be zero A simple procedure to ænd such a set of points is the following: Select any set of n points from E and consider them as column vectors of a matrix While this matrix does not have full rank, add the minimum number of 1 : 11

12 points from E so that the matrix may achieve full rank Continue until a full-rank matrix is obtained, which is guaranteed to happen after selecting at most jej lattice points This gives a set of n independent vectors; picking an additional distinct point produces a simplex In practice it is typically both feasible and as eæcient to just examine the integer lattice points until we ænd n pairs of points such that each pair has vector diæerence equal to è0;:::;0; 1; 0;:::;0è This is, moreover, usually possible within B A shortcut is to ëhide" one of the n variables in the coeæcient æeld This produces an overconstrained system without adding extra polynomial f 0,thus keeping the problem dimension low Our experience with the implementation of this algorithm suggests that hiding a variable is preferable for several systems in robotics and vision ë1ë Formally, we consider the given polynomials as f 1 ;:::;f n Kèx n èëx 1 ;x,1 1 ;:::;x n,1;x,1 n,1 ë and proceed with the construction of M and M 0 as before We can ultimately recover the coordinates of all common zeros as before under the hypothesis that they are isolated and that the value of x n is not repeated between any two roots Since we are free to hide any variable, it suæces that there exist some x i that has geometric multiplicity one for every root Otherwise, we can solve anèn, 1è æ èn, 1è system for every value of the hidden variable Submatrix M 11 which is diagonalized is the largest upper left submatrix created by appropriate row and column permutations, independent ofx n and nonsingular In contrast to the previous case, we do not have a priori knowledge of the sizes of M 11 and M 0, nor is the reduction to an eigenproblem immediate, because M 0 is a matrix polynomial in the hidden variable x n Assume that the highest degree of x n in the given polynomials is d, then M 0 = A d x d n + æææ+ A 1 x n + A 0 ; where the A i are square numeric matrices If A d is nonsingular, the zeros of the systems are recovered from eigenvalue ç and eigenvector v: M 0 èçèv =0, èiç d + A,1 d A d,1ç d,1 + æææ+ A,1 d A 0èv =0 0 I 0 æææ 0, 0 0 æææ 0 I,A,1 d A 0,A,1 d A 1 æææ,a,1 d A d,,a,1 d A d,1 v çv ç d,1 v = ç v çv ç d,1 v A discussion of diæerent strategies for reducing to a generalized eigenproblem when A d is singular is beyond the scope of this paper 8 Mixed Volumes and Minkowski Sums A crucial question in the complexity analysis of these algorithms is the relation between the Mixed Volume and the volume of the Minkowski Sum Q of polytopes Q 1 ;:::;Q n in n-dimensional space Before analyzing complexities, then, we establish some results on the relation of these two quantities We denote by e the basis of the natural logarithm For completeness we start with the result on the class of unmixed systems, ærst shown in ëë Q 1 = æææ= Q n : Lemma 81 For unmixed systems, V èqè =æèe n èmv èq 1 ;:::;Q n è Proof For unmixed systems of polytopes, MV èq 1 ;:::;Q n è=n!v èq 1 è which is, by Stirling's approximation, æèn n =e n èv èq 1 è The Minkowski Sum volume is V èqè =n n V èq 1 è and the claim follows To model mixed systems we have to express their diæerence in shape and volume This is a hard problem in general so we restrict attention to the case where all polytopes have a nonzero n-dimensional volume Deæne the following two objects: the polytope of minimum volume Q ç,1ç ç ç n, such that V èq ç è = minfv èq i è j i =1;:::;ng; : 1

13 and the system's scaling factor s IR, s ç 1 which satisæes the following condition: minimize s : Q i ç sq ç ; i =1;:::;n: The ærst step is a lower bound on the Mixed Volume as a function of a single polytope Lemma 8 If V èq i è é 0 for i =1;:::;n and Q ç is the polytope of minimum volume, then MVèQ 1 ;:::;Q n è ç n! V èq ç è Proof One of the most important inequalities in convexity theory is the Aleksandrov-Fenchel inequality ë1, 1ë which states that MV èq 1 ;:::;Q n è ç MV èq 1 ;Q 1 ;Q ;:::;Q n è MV èq ;Q ;Q ;:::;Q n è; for arbitrary polytopes Q i ç IR n A consequence of this is MV n èq 1 ;:::;Q n è ç èn!è n V èq 1 è æææv èq n è: These results, along with an extensive treatment of the theory, can be found in ë, ë The last inequality implies MV n èq 1 ;:::;Q n è ç èn!è n V n èq ç è which yields the claim since both volume and Mixed Volume are positive-valued functions Theorem 8 Given polytopes Q 1 ;:::;Q n ç IR n such that V èq i è é 0 for all i, deæne Q ç and the system's scaling factor s as above Then V èqè =Oèe n s n èmv èq 1 ;:::;Q n è Proof By deænition, Q ç nx i=1 sq ç = nsq ç ; hence V èqè =ènsè n V èq ç è By the previous lemma, MV èq 1 ;:::;Q n è ç n!v èq ç è Application of Stirling's approximation completes the proof This bound generalizes the unmixed case in which s = 1 Moreover, it is asymptotically quite tight, as seen by the following example Let Q 1 = æææ= Q n,1 ; Q n = sq 1 ; where sé1 and Q ç = Q 1 Then Q =ès + n, 1èQ 1, hence V èqè és n V èq 1 è, and MV èq 1 ;:::;Q n è=sn! V èq 1 è Therefore ç V èqè e n ç s MV èq 1 ;:::;Q n è = æ n,1 ç : n nç For s = n and s = n the lower bound becomes, respectively, æ ç e np s n,ç and æ çe n sn,1 èlog sè n ç : We extend the result to the Minkowski Sum Q 0 of n + 1 polytopes compared with the sum of all n-fold Mixed Volumes D, ie the sum of Mixed Volumes of all subsets of n polytopes Notice that from Deænition, D is the total degree of the sparse resultant Let the scaling factor s of n + 1 polytopes be deæned as the minimum positive real such that Q i ç sq ç, for i =0; 1;:::;n, where Q ç has the minimum volume among all n + 1 polytopes Theorem 8 Given polytopes Q 0 ;Q 1 ;:::;Q n ç IR n, such that V èq i è é 0 for all i, V èq 0 è=oès n e n =nèd, where D is the sum of the n +1n-fold Mixed Volumes and s is this system's scaling factor Proof Q 0 ç sèn+1èq ç hence V èq 0 è ç s n èn+1è n V èq ç è The sum of all n-fold Mixed Volumes is bounded below by the sum of n Mixed Volumes, each on a set of polytopes containing Q ç Then, by Lemma 8, Dénn! V èq ç è therefore V èq ç è=oèe n =n n+1 èd This implies V èq 0 è=oès n e n è1+1=nè n =nèd and the claim follows from lim n!1 è1 + 1=nè n = e 1

14 9 Asymptotic Complexity We have sketched an algorithm for computing monomial bases that consists of testing various edge combinations on whether they lie on the lower envelope of the respective lifted Minkowski Sum or not; the algorithm is described in detail in ë1ë Ignoring the pruning, the algorithm has to test g n combinations, where g is an upper bound on the number of edges in every Newton polytope If r is an upper bound on the number of polytope vertices, g ç r and the number of tests is Oèr n è Note that r is bounded by the maximum number of monomials in any polynomial; the latter provides a diæerent model of sparseness studied in ë0ë Each test is implemented as a Linear Programming question, that decides whether the centroid bp of the lifted cell deæned by n edges lies on the lower envelope of b Q or not: maximize t IR : bp, tz = nx Xr i i=1 j=1 ç ij bv ij ; Xr i j=1 ç ij =1;ç ij ç 0; 8 i =1;:::;n; j =1;:::;r i ; Scalar t expresses the distance between bp and the lower envelope point that lies on the same vertical, so a cell lies on the lower envelope if and only if the optimal value of t is zero Vector z is the unit vector along the èn + 1è-st axis, called the vertical axis Point bv ij is the j-th vertex of lifted polytope b Qi The constraints ensure that bp lies on the same vertical as a variable point deæned as the Minkowski sum of points from the lifted polytopes Linear Programming may be solved by any polynomial-time algorithm; in what follows we apply Karmarkar's algorithm ë18ë The complexity isoèn r èl l + L d èè, where L l is the maximum bit-size of a coordinate in any lifting form l i, i =1;:::;n and L d is the bit-size of the maximum coordinate of any Newton polytope vertex and is bounded by the maximum degree in any variable of an input polynomial Theorem 91 The worst-case bit complexity of our algorithm for computing a monomial basis for the coordinate ring of n polynomials in n variables is r Oènè èlog d,log æè, where r is the maximum number of vertices per polytope and thus bounded by the maximum number of monomials in any polynomial, d is the maximum degree in any variable and æé1 is the probability of failure of the lifting scheme For a constant probability æ and systems with maximum degree d ç roènè the complexity is r Oènè Proof There are r n edge tests at most, each reducing to a Linear Programming application with bit complexity Oèn r èl l + L d èè From è1è, r n =èn Ll è is the probability æ that the lifting fails, then L l = Oèn log r, log æè The general bound is now immediate This is asymptotically optimal because the monomial basis problem is equivalent to Mixed Volume which generalizes Convex Hull Volume which is èp-hard Moreover, it has recently been shown that the Mixed Volume problem is èp-complete ë9ë For most practical applications the extra hypothesis is satisæed and the tighter bound r Oènè applies Now we generalize and formalize the analysis of obtaining resultant matrix M for n + 1 polynomials, based on the results from the previous section and ignoring the polylogarithmic factors in the asymptotic bounds; this is denoted by O æ èæè Again the construction of the mixed subdivision æ 0 æ requires several applications of Linear Programming for which any polynomial-time algorithm may be used; the following bounds were based on Karmarkar's algorithm Lemma 9 ëë Given are n + 1 polynomials in n variables Constructing resultant matrix M has worst-case bit complexity O æ èènrè : jejè, where r is the maximum number of vertices in any Newton polytope and E = èq 0 + æè ë Z n The complexity of explicitly constructing the sparse resultant is bounded byapolynomial in jej and n The cardinality of an integer point set is asymptotically bounded by the volume of their Convex Hull ë10ë, hence jej = OèV èq 0 èè Recall that the scaling factor s of an overconstrained system with Newton polytopes Q 0 ;:::;Q n is the minimum real such that Q i ç sq ç, where Q ç has minimum volume Theorem 9 The construction of resultant matrix M for a system of n +1 polynomials in n variables has worst-case bit complexity O æ ès n e n r : n : Dè, where s is the system's scaling factor, e is the basis of the natural logarithm, r is the maximum number of Newton polytope vertices, bounded by the maximum number of monomials per polynomial, and D is the sum of all n-fold Mixed Volumes 1

15 Proof Theorem 8 implies jej = Oès n e n =nèd and, together with the previous lemma, establishes the claim For typical systems encountered in applications s will be a constant and the algorithm's complexity becomes c Oènè O æ èr : Dè for some constant cé1 Since D is the total degree of the sparse resultant itisalower bound on the algorithm's complexity Passing to the problem of recovering the isolated roots, recall that the initial steps are, given matrix M, to compute matrix M 0 and ænd its eigenvectors We try to ænd n + 1 points in B suæcient for recovering the coordinates of the roots If this is infeasible, there always exist n + 1 points in E that allow us to recover the coordinates through computation of vector v æ of è10è, for each rootæ Let MM èæè be the asymptotic complexity of matrix multiply as a function of the matrix size; currently MMèkè =Oèn : èë9ë It is known that inverting a matrix and computing its determinant and characteristic polynomial all have the same asymptotic complexity as matrix multiply ë0ë The overall bit complexity depends on the bit sizes of the given coeæcients and the root coordinates Let the maximum bit size of these parameters be respectively L c = log c and L æ = log æ, where c and æ are the maximum coeæcient and the maximum root coordinate Then, Lemma 9 Given matrix M, allcommon isolated zeros of polynomials f 1 ;:::;f n are computed with asymptotic algebraic complexity bounded bymmèjejè+mmmènè+oèjejn è, where m is the Mixed Volume The bit complexity is MMèjEjèOèjEj logcè+mmmènèoèn d log æè+oèjejn log dè, where c; æ and d are, respectively, the maximum polynomial coeæcient, root coordinate and polynomial degree in a single variable Proof The matrix operations to compute M 0, eigenvectors v 0 æ and v æ, if necessary, cost MMèjEjè The last two execute on operands of bit size jej logc resulting from the calculation of M 0, hence the ærst term of the overall complexity For each of the m roots, a MMènè operation produces the root coordinates as in the proof of Lemma, assuming that we have found n + 1 aænely independent integer lattice points The operands here are values of E monomials at the roots, hence their maximum bit size is n d log æ for a monomial with every variable raised to nd and hence with total degree n d Enumerating the independent points has worst-case complexity OèjEjn è since it reduces to a rank test on a jej æ n matrix The entries of this matrix are exponent vectors of bit size at most log d Theorem 9 Given is a polynomial system f 1 ;:::;f n in n-variables, deæning a zero-dimensional, radical ideal and let linear polynomial f 0 beasabove Assume that the scaling factor s of the overconstrained system is constant and that the sum of all n-fold Mixed Volumes obeys D = æènmè, where m = MV èf 1 ;:::;f n è Then the worst-case bit complexity of computing all roots of f 1 ;:::;f n is Oènè m log c + mn d log æ + Oènè m log d, where c; d and æ are respectively the maximum polynomial coeæcient, polynomial degree in a single variable and root coordinate Proof We bound MMèkè by k for simplicity and we apply Theorem 8 to the bound in the previous lemma Gríobner bases methods exhibit the same asymptotic complexity, namely single exponential in n and polynomial in m The merit of the sparse elimination methods, though, lies in the fact that their complexity is directly related to the sparseness of the given system and, hence, they are expected to perform better for several problems in practice 10 Open Questions The main open question concerns extending these results to multiple roots, in other words non-radical ideals Suggestions and ideas may originate from current work on the same problem in the context of Gríobner bases ëë An interesting question is to quantify the relation between Mixed Volume and Minkowski Sum volume when polytopes are allowed to have zero n-dimensional volume In this case our lower bound on the Mixed Volume is trivial and we need a diæerent means of expressing the diæerence in shape and volume of the given polytopes For practical applications, an important question is numerical accuracy and conditioning of the matrices This issue deserves separate treatment 1