Ioannis Z. Emiris. On the complexity of sparse elimination. J. Complexity, 12(2):134--166, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....and should allow various further improvements, for instance, by using parallel processing. 2 Reduction of the solution of a polynomial system to matrix eigenproblem In this section we formalize the reduction of the solution of a polynomial system to matrix eigenproblem (cf. 1] 32] 9] [7], 25] 26] We denote by R = C [x1 ; xn ] the ring of polynomials in the variables x = x1 ; xn) with coeOEcients in the eld of complex numbers C . Many of our results are valid for any algebraically closed eld K . To motivate and illustrate the material of this section, we ....

Emiris, I. On the complexity of sparse elimination. J. Complexity 12 (1996), 134166.

....to verify that a guessed answer is correct. Although the current algorithmical practice suggests that computing mixed volumes is harder than computing volumes of polytopes (which is also known as a #P hard problem [20] this is not the case from a complexity point of view as shown in [21] In [25] it is shown that the mixed volume V n (Q) is bounded from below by n vol n (Q ) Q being the polytope of minimum volume in Q. 5.3. Sparse Polynomial Systems solved by Polyhedral Homotopies. The simplest system in the polytope model that still has isolated solutions in (C ) n has exactly ....

I.Z. Emiris. On the complexity of sparse elimination. J. Complexity, 12(2):134--166, 1996.

....when n= 2. It is then easily shown that if one picks a uniformly random a in [ GammaO(m 2 =ffi) O(m 2 =ffi) n , and removes common factors from the coordinates of a, then the probability that a avoids the hyperplanes is at least 1 Gamma ffi. To compute REa , it immediately follows from [Emi96] that this will take O(Sn 9:5 m 6:5 log 2 d ) bit operations for an error probability of , where S is as stated in Lemma 1 and d is an upper bound on the absolute values of the coordinates of the E i . Note that we are also implicitly using the fact that the (n 1) st support consists ....

Emiris, Ioannis Z., "On the Complexity of Sparse Elimination," J. Complexity 12 (1996), no. 2, pp. 134--136.

....inferior asymptotic estimates for the computational complexity of the solution in the worst case (of a dense input) but are preferred by the users for practical implementation. These two approaches rely on computing Groebner bases [KL92] and Newton s polytops (for sparse polynomial systems) [E96], respectively. Then again, in both cases the solution reduces to solv 214 VICTOR Y. PAN ing some polynomial equations in a single variable. Furthermore, the major stage of bounding the step size in the recent successful path following algorithms of [SS93] SS93a] SS93b] and [SS93d] ....

I. Z. EMIRIS, On the complexity of sparse elimination, J. Complexity, 12 (1996), pp. 134--166.

....question in deriving output sensitive upper bounds is the relation between mixed volume and the volume of these Minkowski sums. In manipulating mixed volumes, some fundamental results can be found in [BZ88, Sch93] In particular, the Aleksandrov Fenchel inequality leads to the following bound [Emi96]: MV n (Q 1 ; Qn ) n ) n V (Q 1 ) Delta Delta Delta V (Qn ) For a system of Newton polytopes Q i , dene its scaling factor s to be the minimum real value so that Q i t i ae s Q for all Q i , where Q is the polytope of minimum euclidean volume and the t i 2 R n are ....

....where Q is the polytope of minimum euclidean volume and the t i 2 R n are arbitrary translation vectors. Clearly, s 1 and s is nite if and only if all polytopes have an aOEne span of the same dimension. Let e denote the basis of natural logarithms, and suppose that V (Q i ) 0 for all i. Then [Emi96] V n X i=1 Q i = O(e n s n )MV(Q 1 ; Qn ) V n 1 X i=1 Q i = O e n s n n n 1 X i=1 MV Gammai ; where MV Gammai stands for the mixed volume MV(Q 1 ; Q i Gamma1 ; Q i 1 ; Qn 1 ) i = 1; n 1. 7 2.3.2 The toric resultant In ....

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I.Z. Emiris. On the complexity of sparse elimination. J. Complexity, 12:134166, 1996.

....and their liftings to one higher dimension. It is worthwhile to note that purely combinatorial constructions provide us with important algebraic information. It also provides the basis to solving polynomial equations either by homotopy continuation [VVC94, HS95, VGC96] or by sparse resultants [CE93, Stu94, EC95, Emi96]. However, most existing work has concentrated on solutions with nonzero coordinates. In this paper, we propose an algorithm for dealing with all aOEne solutions, in other words solutions in C d . The limitation to C d 0 is sometimes articial, since in many situations one is interested to know ....

.... computed by several combinatorial methods, in particular [HS95, EC95, VGC96, DGH97] This theory is now entering the mainstream of computational algebra, especially with respect to the fundamental problem in multivariate calculus, namely the computation of all common roots of a polynomial system [CE93, Stu94, VVC94, HS95, EC95, VGC96, Emi96]. The central computation in sparse elimination theory and system solving is nding all mixed cells of the given point conguration. This denes a monomial basis of the coordinate ring and permits computation of the number of roots and numeric approximation of the root vectors. Computing all stable ....

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I.Z. Emiris. On the complexity of sparse elimination. J. Complexity, 12:134166, 1996.

....complexity of algebraic problems in the context of toric elimination and thus provides lower bounds on the complexity of algorithms. In dealing with mixed volumes, some fundamental results can be found in [BZ88, Sch93] In particular, the Aleksandrov Fenchel inequality leads to the following bound [Emi96]: MV(A 1 ; An ) n ) V (A 1 ) Delta Delta Delta V (An ) 1=n : On the other hand, it shall become clear that several toric elimination algorithms rely on some Minkowski sum of Newton polytopes. It turns out that a crucial question in deriving output sensitive upper bounds is the ....

....where A is the polytope of minimum Euclidean volume and the t i 2 R n are arbitrary translation vectors. Clearly, s 1 and s is nite if and only if all polytopes have an aOEne span of the same dimension. Let e denote the basis of natural logarithms, and suppose that V (A i ) 0 for all i. Then [Emi96] V n X i=1 A i = O(e n s n )MV(A 1 ; An ) V n 1 X i=1 A i = O e n s n n n 1 X i=1 MV Gammai ; where MV Gammai stands for the mixed volume MV(A 1 ; A i Gamma1 ; A i 1 ; An 1 ) i = 1; n 1. The toric resultant For a ....

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I.Z. Emiris. On the complexity of sparse elimination. J. Complexity, 12:134166, 1996.

....of a homogeneous GCP. In the sparse context, the only existing approach focuses on the u resultant and requires substantial additional computation [Roj99] Recall that the u resultant reduces the solution of systems of n equations to a determinant factorization by introducing a new polynomial [Emi96, CLO98]. We settle the sparse case in its generality by an e cient generalization of the GCP. The main contribution of this paper is a simple perturbation scheme that produces sparse projection operators de ned by a resultant matrix determinant and which, after specialization, are non identically zero ....

....f i is MV(f 0 ; f i 1 ; f i 1 ; fn ) which expresses the generic number of toric roots of this subsystem. MV( is the mixed volume of the respective Newton polytopes, where the Newton polytope of a polynomial is the convex hull of the exponent vectors in the polynomial s support [GKZ94, Stu94, Emi96, CLO98]. In [Can90] a linear perturbation called Generalized Characteristic Polynomial (GCP) allows us to compute a projection operator which is not identically zero, but vanishes on all proper components of the system f i = 0 in the dense case, i.e. when A i = fx 2 N n 0 : x 1 : xn d i g. ....

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I.Z. Emiris. On the complexity of sparse elimination. J. Complexity, 12:134-166, 1996.

....of algebraic problems in the context of toric elimination and thus provides lower bounds on the complexity of algorithms. In dealing with mixed volumes, some fundamental results can be found in [BZ88, Sch93] In particular, the Aleksandrov Fenchel inequality leads to the following bound [Emi96] MV(Q 1 ; Qn ) n ) V (Q 1 ) Delta Delta Delta V (Qn ) 1=n : On the other hand, we will see that the Minkowski sum Q 1 : Qn contains all the required information to study and solve a system of n generic polynomials. In the forthcoming sections, it shall become clear that ....

....Q is the polytope of minimum Euclidean volume and the t i 2 R n are arbitrary translation vectors. Clearly, s 1 and s is nite if and only if all polytopes have an aOEne span of the same dimension. Let e denote the basis of natural logarithms, and suppose that V (Q i ) 0 for all i. Then [Emi96] V n X i=1 Q i = O(e n s n )MV(Q 1 ; Qn ) V n 1 X i=1 Q i = O e n s n n n 1 X i=1 MV Gammai ; where MV Gammai stands for the mixed volume MV(Q 1 ; Q i Gamma1 ; Q i 1 ; Qn 1 ) i = 1; n 1. The toric resultant In toric ....

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I.Z. Emiris. On the complexity of sparse elimination. J. Complexity, 12:134166, 1996.

....Toeplitz structure [EP97] For solving systems of n polynomial equations in n unknowns, certain properties of the Newton matrix must be established. One approach is that of [PS96] whereas a simpler proof, based on the present algorithm, which reduces root nding to an eigenproblem is found in [Emi96]. This result extends older work in [AS88, MC93] Observe that the exact sparse resultant is not necessary, since a Newton matrix suOEces. The computation of solution sets with positive dimension has been undertaken in [KM95] In practice, questions of degeneracy may have to be addressed, as in ....

....question in the complexity analysis of sparse elimination algorithms is the relation between mixed volume and the volume of the Minkowski sum Q = Q 1 ; Qn 1 ae R n . We denote by e 2:718 the exponential base. A weaker version of the following result rst appeared in [CE93] Lemma 3. 8 [Emi96] For unmixed systems with Newton polytopes Q 1 = Delta Delta Delta = Qn 1 , Vol(Q 1 Delta Delta Delta Qn 1 ) Theta e n deg R n 3=2 ; where deg R is the total degree of the sparse resultant in the input coeOEcients. To model mixed systems we have to express their dioeerence ....

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I.Z. Emiris. On the complexity of sparse elimination. J. Complexity, 12:134166, 1996.

....Universiteit Leuven, Department of Computer Science, Celestijnenlaan 200 A, B 3001 Heverlee, Belgium. E mail: jan.verschelde na net.ornl.gov 21 information. It also provides the basis to solving polynomial equations either by homotopy continuation [VVC94, HS95, VGC96] or by sparse resultants [CE93, Stu94, EC95, Emi96]. However, most existing work has concentrated on solutions with nonzero coordinates. In this paper, we propose an algorithm for dealing with all aOEne solutions, in other words solutions in C d . The limitation to C d 0 is sometimes articial, since in many situations one is interested to ....

.... computed by several combinatorial methods, in particular [HS95, EC95, VGC96, DGH97] This theory is now entering the mainstream of computational algebra, especially with respect to the fundamental problem in multivariate calculus, namely the computation of all common roots of a polynomial system [CE93, Stu94, VVC94, HS95, EC95, VGC96, Emi96]. The central computation in sparse elimination theory and system solving is nding all mixed cells of the given point conguration. This denes a monomial basis of the coordinate ring and permits computation of the number of roots and numeric approximation of the root vectors. Computing all stable ....

[Article contains additional citation context not shown here]

I.Z. Emiris. On the complexity of sparse elimination. J. Complexity, 12:134166, 1996.

.... are robust to input perturbations, and have lower worst case complexity than Gr#bner bases, which is the best established and most general method today [CLO98] Exploiting structure is achieved in a strong and predictable way by the theory of sparse elimination; see, e.g. CLO98, EC95, Emi96, SZ94] and the next two sections. This theory has generalized several results of classical variable elimination theory. The model of sparsity is a combinatorial one, and raises several problems in general dimensional convex geometry. One bottleneck is due to the computational question examined in ....

....so that, with high probability, the solution to the subproblem satises a large number of constraints. 3 Sparse elimination theory This section sketches the theory of sparse variable elimination and the main combinatorial concepts required. Further information can be found in [CLO98, EC95, Emi96, SZ94] Sparse elimination allows us to consider Laurent polynomials f 2 K[x 1 ; x Gamma1 1 ; xn ; x Gamma1 n ] where K is an arbitrary eld of coeOEcients. Denition 3.1 The support of polynomial f 2 K[x 1 ; x Gamma1 1 ; xn ; x Gamma1 n ] is the set of exponent ....

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I.Z. Emiris. On the complexity of sparse elimination. J. Complexity, 12:134166, 1996.

....equations. When all roots are isolated, methods based on resultant matrices have lower complexity, are robust to input perturbations, and can strongly model the sparse structure of the input polynomials. The latter is achieved in a strong and predictable way by the theory of sparse elimination [4, 5, 6, 14]. This theory has generalized several results of classical variable elimination theory. The model of sparsity is a combinatorial one, and raises several problems in general dimensional convex geometry. The hardest computational problem encountered is the one examined in this paper: Problem ....

....section 6. Counting the number of integer points in an integer polytope is #P complete when the dimension is not xed [8] 3 Sparse elimination theory This section sketches the theory of sparse variable elimination and the main combinatorial concepts required. Further information can be found in [4, 5, 6, 14]. Sparse elimination allows us to consider Laurent polynomials f 2 K[x 1 ; x Gamma1 1 ; x n ; x Gamma1 n ] denoted K[x; x Gamma1 ] where K is an arbitrary eld of coeOEcients. Denition 3.1 The support supp(f) ae Z n of polynomial f 2 K[x; x Gamma1 ] is the set of exponent ....

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I.Z. Emiris. On the complexity of sparse elimination. J. Complexity, 12:134166, 1996.

....matrix. The emphasis here is placed on numerical issues and the application of our implementation to concrete problems. We describe very brieAEy the main steps in sparse elimination and the construction of sparse resultant matrices. Most proofs are omitted but can be found in [CE93, Emi94, EC95, Emi96] The study of coordinate rings of varieties in K n , where K is a eld, has been shown to be particularly useful in studying systems of polynomial equations. We concentrate on zero dimensional varieties for which it is known that the coordinate ring forms a nite dimensional vector space and, ....

....eld K. We are interested in the case that V has dimension zero. Then, its coordinate ring K[x; x Gamma1 ] I is an m dimensional vector space over K by theorem 2. 4, where m = MV (f 1 ; f n ) MV (Q 1 ; Q n ) Using the subdivision based construction it is easy to show [ER94, Emi96] that generically a monomial basis of K[x; x Gamma1 ] I can be found among the monomials of Q 1 Delta Delta Delta Q n ae R n : Moreover, resultant matrix M produces the multiplication map for any given f 0 . This is a matrix, or an endomorphism, that serves in computing in the ....

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I.Z. Emiris. On the complexity of sparse elimination. J. Complexity, 12:134166, 1996.

.... Delta Delta n in V ( 1 A 1 Delta Delta Delta nAn ) expanded as a polynomial in 1 ; n . A complete account of sparse elimination, including equivalent de nitions for mixed volume in general dimension an an eOEcient algorithm for its computation, can be found in [PS93, EC95, Emi96] and their references. See section 5 for the location at which the code is publicly available. In our case the polytopes C i are squares of size 2. Bernstein s theorem bounds the number of common roots with no zero coordinates by the mixed volume of the Newton polytopes. The mixed volume of the ....

....sometimes this problem is regarded as the generalized eigenproblem or the problem of computing the matrix kernel vectors. It is not a new result in computational algebra, but it has not widely been used for practical purposes; the interested reader may consult any of [KL92, EC95, NR95, Emi96, CM96, CLO97] An important feature of our method is precisely that it reduces to matrix operations for which powerful and accurate implementations already exist in the public domain, such as [ABB 95] which is used in the sparse resultant solver implemented in C [Emi97] and in ALP; see ....

[Article contains additional citation context not shown here]

I.Z. Emiris. On the complexity of sparse elimination. J. Complexity, 12:134166, 1996.

....space complexity bounds for their construction, the computation of the sparse resultant and the solution of polynomial systems. Construction and manipulation of Newton matrices is a critical operation in some of the most eOEcient known algorithms for solving zero dimensional systems of equations [3, 12, 5, 13]. Our practical motivation is the real time solution of systems with, say, up to 10 variables; or the computation of the resultant polynomial, for instance in graphics and modeling applications where the implicit expression of a curve or surface is precisely the resultant. Such systems may give ....

....had to be omitted here because of space restrictions. Recent interest in matrix based methods is supported by certain practical results that have established resultants, along with Gr#bner bases and continuation techniques, as the method of choice in solving zero dimensional polynomial systems [17, 3, 12, 5, 13]. A generalization of the classical resultant was introduced in the context of sparse elimination theory (outlined in section 5) Two main algorithms, generalizing Sylvester s as well as Macaulay s constructions, have been proposed for constructing Newton, or sparse resultant, matrices: The ....

[Article contains additional citation context not shown here]

Emiris, I. On the complexity of sparse elimination. J. Complexity 12 (1996), 134166.

....and their liftings to one higher dimension. It is worthwhile to note that purely combinatorial constructions provide us with important algebraic information. It also provides the basis to solving polynomial equations either by homotopy continuation [VVC94, HS95, VGC96] or by sparse resultants [CE93, Stu94, EC95, Emi96]. However, most existing work has concentrated on solutions with nonzero coordinates. Work partially supported by Esprit LTR project no. 21.024 FRISCO. This work was conducted when the second author was post doctor at the K.U.Leuven, Belgium. y INRIA Sophia Antipolis, 2004 Route des Lucioles ....

.... computed by several combinatorial methods, in particular [HS95, EC95, VGC96, DGH98] This theory is now entering the mainstream of computational algebra, especially with respect to the fundamental problem in multivariate calculus, namely the computation of all common roots of a polynomial system [CE93, Stu94, VVC94, HS95, EC95, VGC96, Emi96]. The central computation in sparse elimination theory and system solving is finding all mixed cells of the given point configuration. This defines a monomial basis of the coordinate ring and permits computation of the number of roots and numeric approximation of the root vectors. Computing all ....

[Article contains additional citation context not shown here]

I.Z. Emiris. On the complexity of sparse elimination. J. Complexity, 12:134--166, 1996.

....bounds and exploit sparseness. This theory has close links with combinatorial geometry. Polynomials are speci ed by their support and its convex hull. Sparse elimination de nes the sparse resultant, whose degree depends on these convex polytopes instead of the total degrees [CLO97, GKZ92, Stu94, Emi96] Constructing matrices whose determinant is a nontrivial multiple of the sparse resultant involves algebraic and geometric computation and yields matrices that generalize those of Sylvester and Macaulay [CLO97, Stu94, Emi96, CE93, EC95] The second branch of resultant matrix constructions stems ....

.... these convex polytopes instead of the total degrees [CLO97, GKZ92, Stu94, Emi96] Constructing matrices whose determinant is a nontrivial multiple of the sparse resultant involves algebraic and geometric computation and yields matrices that generalize those of Sylvester and Macaulay [CLO97, Stu94, Emi96, CE93, EC95] The second branch of resultant matrix constructions stems from B#zout s method for the resultant of two univariate polynomials. For the example system in (1) the resultant matrix is a 0 b 1 Gamma a 1 b 0 a 0 b 2 a 0 a 1 : 2) Notice that both matrices in (1) and (2) have ....

[Article contains additional citation context not shown here]

I.Z. Emiris. On the complexity of sparse elimination. J. Complexity, 12:134166, 1996.

....linear algebra; sometimes this problem is regarded as the generalized eigenproblem or the problem of computing the matrix kernel vectors. It is not a new result in computational algebra, but it has not widely been used for practical purposes; the interested reader may consult any of [EC95, CM96, Emi96] An important feature INRIA Polynomial system solving: the case of a six atom molecule 11 of our method is precisely that it reduces to matrix operations for which powerful and accurate implementations already exist in the public domain, such as [ABB 92] which is used in the sparse ....

.... for constructing sparse resultant matrices (also called Newton matrices) whose determinant is a nontrivial multiple of the sparse resultant and which express multiplication in the quotient ring, can be found in [CE93, CP93, EC95] The size of the Newton matrices scales with mixed volume [Emi96] Linear algebra methods for reducing the matrix size and removing some genericity requirements, can be found in [Emi94] The second class of methods relies on the theory of residues, and does not assume that the system is generic,but only that it is a complete intersection (the codimension of ....

I.Z. Emiris. On the complexity of sparse elimination. J. Complexity, 12:134166, 1996.

....proposing algorithms and analyzing their complexities, then, we establish some results on the relation of these two quantities. We extend the discussion to systems of n 1 polytopes where we establish bounds in terms of the sum of n fold mixed volumes. Most of these results first appeared in [Emi94] We denote by e the exponential base. Lemma 2.2.1 For unmixed systems Q 1 = Delta Delta Delta = Q n . Then Vol(Q) Theta (e n = p n)MV (Q 1 ; Q n ) Proof Stirling s approximation is used throughout the thesis to provide asymptotic bounds: n = Theta ( p 2 nn n e n ) ....

....obtain smaller matrices. Polynomials need not be homogenized and the explicit resultant polynomial is never computed. Instead a matrix formula is used to reduce root finding to an eigenproblem, which allows the application of powerful techniques from linear algebra. We generalize the results in [Emi94] to minimize the a priori knowledge required by the algorithm, thus incorporating both resultant matrix construction algorithms. The study of coordinate rings of varieties in K n , where K is a field, has been shown to be particularly useful in studying systems of polynomial equations. For a ....

I.Z. Emiris. On the complexity of sparse elimination. Technical Report 840, Computer Science Division, U.C. Berkeley, 1994. Submitted for publication.

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Emi96 I.Z. Emiris. On the complexity of sparse elimination. J. Complexity, 12:134166, 1996.

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Ioannis Z. Emiris. On the complexity of sparse elimination. J. Complexity, 12(2):134--166, 1996.

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I. Emiris. On the Complexity of Sparse Elimination. J. Complexity , vol. 12:pp. 134--166, 1996.

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I.Z. Emiris. On the complexity of sparse elimination. J. Complexity, 12:134--166, 1996.

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