B. Sturmfels. Sparse elimination theory. In Computational Algebraic Geometry and Commutative Algebra. Cambridge University Press, June 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....construction follows the same process as in the previous section, except that the notion of degree is changed. We consider n 1 Laurent polynomials f 0 ; f n 2 L = K [t 1 ; t n ] and we replace the constrains on the degree by constrains on the support of the polynomials [42, 85, 35]: Let x a polytope A i Z and assume that the support of f i is in A i : f i = 2A i c i; t The support of p = c x is the set of 2 Z such that c 6= 0 20 We denote by A the Minkowski sum of these polytopes (A = A 0 A n ) to which we associate the toric ....

B. Sturmfels, Sparse elimination theory, in Computational Algebraic Geometry and Commutative Algebra, D. Eisenbud and L. Robianno, eds., Cambridge, Cambridge Univ. Press, 1993, pp. 264-298.

....is a simple fact, made precise in Lemma 2.1 below, that any generic lifting function on a polyhedral complex induces a simplicial subdivision. This fact is used frequently in applications of subdivisions to the computation of mixed volumes, polyhedral homotopies, and toric (or sparse) resultants [Stu93, HS95, CE00, Roj00]. The last two constructions give e ective recent techniques, sometimes more ecient than Gr obner bases, for solving systems of polynomial equations. However, it should be emphasized that the lifting functions considered here and in [KKMS73] are more general than those in [Stu93, HS95, CE00] the ....

....HS95, CE00, Roj00] The last two constructions give e ective recent techniques, sometimes more ecient than Gr obner bases, for solving systems of polynomial equations. However, it should be emphasized that the lifting functions considered here and in [KKMS73] are more general than those in [Stu93, HS95, CE00]: the lifting functions in the latter references are completely determined by the values assigned to the vertices of . We will call these more restricted lifting functions verticial. The verticial lifting functions are a bit more economical in the sense that their corresponding subdivisions ....

Sturmfels, Bernd, \Sparse Elimination Theory," In D. Eisenbud and L. Robbiano, editors, Proc. Computat. Algebraic Geom. and Commut. Algebra

....known approaches under the same terminology of Sylvester map. In particular, we will cover the cases where X = P n is the projective space of dimension n, which yields the classical resultant (see [15] 28] and where X is a toric variety, which yields the so called toric resultant (see [9] [26], 2] The resultant can be computed as a divisor of the determinant of a map, which generalizes the Sylvester map for two polynomials in one variable. Let V 0 ; Vn be the n 1 vector spaces generated by monomials x E i = fx ff ; ff 2 E i g, where E i is the set of the exponents, E ....

....direction ffi 2 Q n . For any polytope C, let C ffi denote the polytope obtained from C by removing its facets whose normals have positive 9 inner products with ffi. Taking E i ( P j 6=i C j ) ffi and F = P j C j ) ffi allows us to dene the desired map S. We refer the reader to [9] [26], 2] for further details. Now, let us check, step by step, that hypotheses 3.1 are satised. In the experiments (cf. section 5) we choose a linear form for f 0 . Here, we only assume that f 0 contains a constant term. As all the monomials of f 0 x E0 are in V , this implies that the set of the ....

B. Sturmfels. Sparse elimination theory. In D. Eisenbud and L. Robianno, editors, Computational Algebraic Geometry and Commutative Algebra, pages 264298. Cambridge Univ. Press,

....the open subset U is (K f0g) m and the i;j are (Laurent) monomials in t 1 1 ; t 1 m . This yields to the notion of toric resultant, which is a condition on c such that the system f c homogenized in a convenient way has a solution in the corresponding toric variety [GKZ94] [Stu93], CE93] Cox95] A resultant over a unirational algebraic variety is constructed in [BEM00] If X is a projective variety parameterized by a map de ned on an open subset U A m , and i;j are homogeneous polynomials such that i;j = i;j . The existence of an irreducible resultant ....

B. Sturmfels. Sparse elimination theory. In D. Eisenbud and L. Robbiano, editors, Proc. Computat. Algebraic Geom. and Commut. Algebra

....same terminology of Sylvester map, several known approaches. In particular, we will cover the cases where X = P n is the projective space of dimension n, which yields the classical resultant (see [18] 34] and where X is a toric variety, which yields the so called toric resultant (see [12] [33], 3] These resultants can be computed as a factor of the determinant of a map, which generalizes the Sylvester map for two polynomials in one variable. Let V0 ; Vn be n 1 vector spaces generated by monomials x E i = fx ff ; ff 2 E i g, where E i is the set of the exponents, E i = ....

....ffi 2 Q n . For any polytope C, let C ffi denote the polytope obtained from C, by removing its facets whose normals have positive inner products with ffi. Taking E i = P j 6=i C j ) ffi and F = P j C j ) ffi , allows to dene the expected map S. We refer the reader to [12] [33], 3] for further details. Now let us check step by step, that hypotheses 3.2 are satised. In the examples, we will choose a linear form, for f0 . Here, we only assume that f0 contains a constant term. As all the monomials of f0 x E 0 are in V, it implies that the set of the monomials x ....

Sturmfels, B. Sparse elimination theory. In Computational Algebraic Geometry and Commutative Algebra (

....the system has a solution, arise in many application domains including the ones listed above. For an overview as well as details, the reader may consult [Hof89, DKM92, Emi94] Three major multivariate resultant formulations are the Macaulay [Mac16, Can90] Dixon [Dix08, KSY94, KS95] and sparse [Stu91, CE93, Emi94] resultant formulations. Given a polynomial system, these formulations construct matrices, called the Macaulay matrix, the Dixon matrix and the sparse resultant matrix, respectively. In the Macaulay formulation, the ratio of the determinants of the Macaulay matrix and one of its sub matrices gives ....

B. Sturmfels. Sparse elimination theory. In D. Eisenbud and eds. L. Robbiano, editors, Proc. Computat. Algebraic Geom. and Commut. Algebra, Cortona,Italy, June 1991. Cambridge Univ. Press. 14

.... systems and computing a projection operator from which the resultant can be extracted [12] This method has been experimentally found to be superior in performance on a wide variety of examples, in comparison with other elimination methods including Macaulay resultants, sparse resultants [3, 14, 8], the characteristic set construction [4] and the Gr obnerbasis construction [1, 2] The method takes less time, less space, as well as the extraneous factors seem to be fewer (except in the case of the Gr obnerbasis method which gives the exact resultant) 9] We also proved that for the unmixed ....

Sturmfels, B. Sparse elimination theory. In Proc. Computat. Algebraic Geom. and Commut. Algebra (Cortona,Italy, June 1991), D. Eisenbud and e. L. Robbiano, Eds., Cambridge Univ. Press.

....gure 1) to our example, we see that the only w we need worry about in condition (a 2 ) are (in counter clockwise order) 1; 2) 1; 1) 3; 2) 1; 2) and (3; 2) Furthermore, the corresponding sparse resultants are easily seen to be 1 , 1 , 8 , 5 8 6 3 , and 3 . The papers [Stu92, Stu93, PS93, Stu94, Stu98] and the book [GKZ94] contain some very nice examples of how to compute low dimensional sparse resultants. Condition (b 2 ) then clearly specializes to two one dimensional cases of condition (a 2 ) More conservatively, the fundamental theorem of algebra could also be applied to (b 2 ) So it ....

....CE=C has a root in (K ) n , then ResE (C) 0. For xed E, the polynomial ResE ( can then be de ned (up to a nonzero scalar multiple) as the unique polynomial in CE of least total degree satisfying this last property. The computation of ResE ( is a deep subject and we refer the reader to [GKZ90, Stu93, PS93, CE93, SZ94, Stu94, GKZ94, EC95, Stu98] for further background on sparse resultants. For convenience, we will use ResE (F ) in place of ResE (C) whenever the coecients of F have been specialized to some C 2 K jEj . We also point out the following important fact: ResE (F ) 0 does not necessarily imply that F has a root in (K ) ....

Sturmfels, Bernd, \Sparse Elimination Theory," In D. Eisenbud and L. Robbiano, editors, Proc. Computat. Algebraic Geom. and Commut. Algebra

....to be faster than the GCP for sparse polynomial systems, the toric GCP appears to be far more competitive in such a comparison. From a more theoretical point of view, our focus on univariate reduction is a useful addition to Sturmfels foundational work on sparse elimination theory in the large [Stu93]. We also point out that [Stu93] provides a wonderfully clear introduction to some of the toric variety techniques we refer to. Going further into the intersection of algebraic geometry and complexity theory, there is a beautiful new approach to elimination theory founded by the school of ....

....polynomial systems, the toric GCP appears to be far more competitive in such a comparison. From a more theoretical point of view, our focus on univariate reduction is a useful addition to Sturmfels foundational work on sparse elimination theory in the large [Stu93] We also point out that [Stu93] provides a wonderfully clear introduction to some of the toric variety techniques we refer to. Going further into the intersection of algebraic geometry and complexity theory, there is a beautiful new approach to elimination theory founded by the school of Heintz, et al. PK95, GHMP95] Our ....

[Article contains additional citation context not shown here]

Sturmfels, Bernd, \Sparse Elimination Theory," In D. Eisenbud and L. Robbiano, editors, Proc. Computat. Algebraic Geom. and Commut. Algebra

....ideal, i.e. generated by monomials. Then by means of Buchberger s algorithm (cf. 3] 15.3) one may compute from a given set of generators of I a set of generators of in c (I) and from that the prime ideals of maximal dimension associated to in c (I) together with their multiplicities (cf. [18], Proposition 3.4 or Lemma 4.3 of the present paper) For a subset W of 0, N , let PW denote the ideal of polynomials vanishing identically on the coordinate plane given by x j = 0 for j # W . Then the prime ideals of maximal dimension associated to in c (I) are PW 1 , PWg ....

....of sets W # 0, N with the property that for every i # 1, T there is a j # W with a ij 0. Given W # S(I) let AW (I) a # Z N 1 #0 : supp a # W, a ## a i,W for all i = 1, T . 4. 2) We have included a proof of the following simple lemma (see also [18], Proposition 3.4) DIOPHANTINE INEQUALITIES ON PROJECTIVE VARIETIES 15 Lemma 4.3. Let W 1 , W g be the non empty sets in S(I) of minimal cardinality. Then PW 1 , PWg are the prime ideals of maximal dimension associated to I. Further, for i = 1, g, the multiplicity PW i ....

B. Sturmfels, Sparse elimination theory, in Computational Algebraic Geometry and Commutative Algebra, ed. D. Eisenbud and L. Robbiano, (1993) 264---298.

....known approaches under the same terminology of Sylvester map. In particular, we will cover the cases where X = P n is the projective space of dimension n, which yields the classical resultant (see [15] 28] and where X is a toric variety, which yields the so called toric resultant (see [9] [26], 2] The resultant can be computed as a divisor of the determinant of a map, which generalizes the Sylvester map for two polynomials in one variable. Let V 0 ; Vn be the n 1 vector spaces generated by monomials x E i = fx ff ; ff 2 E i g, where E i is the set of the exponents, E ....

....direction ffi 2 Q n . For any polytope C, let C ffi denote the polytope obtained from C by removing its facets whose normals have positive inner products with ffi. Taking E i = P j 6=i C j ) ffi and F = P j C j ) ffi allows us to dene the desired map S. We refer the reader to [9] [26], 2] for further details. Now, let us check, step by step, that hypotheses 3.1 are satised. In the experiments (cf. section 5) we choose a linear form for f 0 . Here, we only assume that f 0 contains a constant term. As all the monomials of f 0 x E0 are in V , this implies that the set of the ....

B. Sturmfels. Sparse elimination theory. In D. Eisenbud and L. Robianno, editors, Computational Algebraic Geometry and Commutative Algebra, pages 264298. Cambridge Univ. Press,

.... these last decades have witnessed a renewal of elimination theory (Jouanolou, 1991) Gelfand et al. 1994) Eisenbud, 1994) partly motivated by applications in eoeective algebraic geometry and more specially in polynomial system solving (Chistov, 1986) Grigoryev, 1986) Pedersen and Sturmfels, 1993), Manocha and Canny, 1993) Chardin, 1995) Mourrain, 1998) Indeed many operations used in this domain involve projections of varieties and elimination of variables. Resultant constructions yield a direct answer to such problems. After a preprocessing step of the polynomial equations, one ....

.... the notion of resultant to more general varieties than the projective space (Gelfand et al. 1994) The recent eoeorts in this direction concern resultants over toric varieties, and more precisely explicit matrix constructions whose determinant is a nontrivial multiple of the toric resultant (Sturmfels, 1993), Gelfand et al. 1994) chap. 8] Canny and Emiris, 1993) In this work, we aim at extending such constructions to resultants over general varieties. We propose a systematic method based on Bezoutian matrices, which yields a nontrivial multiple of the resultant over a projective variety X , when ....

Sturmfels, B. (1993). Sparse elimination theory. In Eisenbud, D., Robianno, L., editors, Computational Algebraic Geometry and Commutative Algebra, pages 264298. Cambridge Univ. Press. (Proc.

....section, we review the definition and some elementary properties of multivariate resultants. For details, see (Macaulay 1916, Cayley 1865, Dixon 1908, White 1909, der Waerden 1966) For various modern methods for computing resultants, see (Kaltofen and Lakshman 1988, Zippel 1990, Chardin 1991, Sturmfels 1991, Canny and Imiris 1993, Manocha 1993, Kapur and Saxena 1995) Also the recent survey (Kapur and Lakshman 1992) gives highly readable descriptions of both classical and modern methods. We begin by fixing some notations that will be used throughout this paper. Notations 2.1. D an integral ....

Sturmfels, B. (1991). Sparse elimination theory. In Robbiano, L., Eisenbud, D., editors, Proc. Cortona 1991. Cambridge University Press.

....of the resultant (called a projection operator) as a determinant. There are three major methods to compute the resultant or a projection operator the classical formulations by Dixon [4] and Macaulay [12] developed early this century, and the recently developed sparse resultant formulation [11, 6]. All three methods construct matrices whose determinant is either the resultant, or a projection operator. The Macaulay formulation uses the traditional Bezout bound on the number of solutions of a polynomial system to construct the Macaulay matrix. The size of this matrix depends on the total ....

....found to be more efficient than the Macaulay and the sparse resultant formulations [9, 10] However, being a classical approach (developed in 1908) the relationship between the Dixon formulation and the modern BKK bounds, or the structure of the input polytopes, is not well understood. In [11], Sturmfels gave the bracket representation of the Dixon resultant for bi homogeneous systems; however the Dixon resultant was not analyzed in its full generality. Moreover, it has been unclear whether the Dixon formulation exploits the sparsity of input polynomials, and if so, to what extent ....

Sturmfels B., Sparse Elimination Theory, Proc. Computat. Algebraic Geom. and Commut. Algebra, D. Eisenbud and L. Robbiano, eds., Cortona , Italy, June 1991.

....and solid modeling are concerned, the use of resultants was resurrected by Sederberg for implicitizing parametric curves and surfaces [Sed83] In this paper we will be dealing with resultants of two polynomials in one unknown. Surveys on various formulations of resultants are given in [Sed83, Stu91] and effective techniques for computing and applying them are presented in [Man92] Given two polynomials in one unknown, their resultant is a polynomial in their coefficients. Moreover, the vanishing of the resultant is a necessary and sufficient condition for the two polynomials to have a ....

B. Sturmfels. Sparse elimination theory. In D. Eisenbud and L. Robbiano, editors, Computational Algebraic Geometry and Commutative Algebra. Cambridge University Press, 1991.

....thus leading to the notion of the multivariate resultant. The three most commonly used multivariate resultant formulations are the Dixon [Dixon 1908, Kapur and Saxena 1995] Macaulay [Macaulay 1916, Canny 1990, Kaltofen and Lakshman 1988] and sparse resultant formulations [Canny and Emiris 1993, Sturmfels 1991]. The theory of Grobner bases provides powerful tools for performing computations in multivariate polynomial rings. Formulating the problem of solving systems of polynomial equations in terms of polynomial ideals, we will see that a Grobner basis can be computed from the input polynomial set, thus ....

....variables vanishes exactly when there exists a common solution in projective space [van der Waerden 1950, Kapur and Lakshman Y. N. 1992] The sparse resultant characterizes solvability over a smaller space which coincides with affine space under certain genericity conditions [Gelfand et al. 1994, Sturmfels 1991]. The main algorithmic question, then, is to construct a matrix whose determinant is the resultant or a non trivial multiple of it. Due to the special structure of the Sylvester matrix, B ezout developed a method for computing the resultant as a determinant of order Max(m;n) during the eighteenth ....

[Article contains additional citation context not shown here]

B. Sturmfels. Sparse elimination theory. In D. Eisenbud and L. Robbiano, editors, Proc. Computat. Algebraic Geom. and Commut. Algebra, Cortona, Italy, 1991.

....the resultant as a function of more than one determinants or give a single matrix whose determinant is a nontrivial multiple of the resultant. In this paper, we concentrate on such approaches. Three major multivariate resultant formulations are the Macaulay [20, 4] Dixon [10, 18] and sparse [22, 5] resultant formulations. Given a set of polynomials, these formulations construct matrices, called the Macaulay matrix, the Dixon matrix and the sparse resultant matrix, respectively. In Macaulay formulation, the ratio of the determinants of the Macaulay matrix and one of its submatrices gives the ....

....of computing the determinant. Even if it is square, it may be singular, resulting in a trivial projection operator. 2. 4 The Sparse Resultant Formulation This approach is based on the recent results pertaining to sparse polynomial systems (Bernshtein [2] Gelfand et al. 16] and Sturmfels [22]) Sparse resultants appeared in Sturmfels Zelevinsky [23] its matrix formulation was given by Canny Emiris [5] and was successively improved in [6, 11, 12] using better heuristics. The Bezout bound on the 4 n 1 nonhomogeneous polynomials p1 ; pn 1 in x1 ; xn are called ....

Sturmfels B., Sparse Elimination Theory, Proc. Computat. Algebraic Geom. and Commut. Algebra, D. Eisenbud and L. Robbiano, eds., Cortona, Italy, June 1991.

....of the newton polytopes corresponding to each equation. Bernstein showed that his bound is exact if all the coefficients of the polynomial system are generic [Ber75] Resultant formulations in terms of matrices and determinants based on Bernstein bound have appeared in the literature as well [Stu91, SZ94, CE93] Given a system of polynomial equations, sparse or dense, it is possible to express their resultant in terms of matrices and determinants. We use this linear algebra formulation in the algorithms presented in the following sections. 2.2 Matrix Computations In this section we ....

B. Sturmfels. Sparse elimination theory. In D. Eisenbud and L. Robbiano, editors, Computational Algebraic Geometry and Commutative Algebra. Cambridge University Press, 1991.

....the algebraic complexity of the system. For dense polynomial systems, it is the Bezout bound corresponding to the product of the degrees of the equation; and for sparse polynomial systems, it is the BKK bound corresponding to the mixed volume of the Newton polytope corresponding to each equations [7, 45]. Homotopy methods use the solutions of a known system along with tracing algorithms to find the solutions of the given system. The tracing steps corresponds to Newton s iteration. 3 Collision Detection for Polyhedra In this section, we summarize a simple and efficient collision detection ....

B. Sturmfels. Sparse elimination theory. In D. Eisenbud and L. Robbiano, editors, Computational Algebraic Geometry and Commutative Algebra. Cambridge University Press, 1991.

.... systems and computing a projection operator from which the resultant can be extracted [11] This method has been experimentally found to be superior in performance on a wide variety of examples, in comparison with other elimination methods including Macaulay resultants, sparse resultants [3, 17, 7], the characteristic set construction [5] and the Grobner basis construction [1, 2] The method takes less time, less space, and seems to generate fewer extraneous factors [10] except in the case of the Grobner basis method which gives the exact resultant) Recently, we have also been able to ....

Sturmfels B., Sparse Elimination Theory, Proc. Computat. Algebraic Geom. and Commut. Algebra, D. Eisenbud and L. Robbiano, eds., Cortona,Italy, Cambridge Univ. Press, June 1991.

....efficient to eliminate all variables together from a set of polynomials, thus leading to the notion of the multivariate resultant. The three most commonly used multivariate resultant formulations are the Dixon [Dix08, KS95b] Macaulay [Mac16, Can90, KL88] and sparse resultant formulations [CE93a, Stu91] The theory of Grobner bases provides powerful tools for performing computations in multivariate polynomial rings. Formulating the problem of solving systems of polynomial equations in terms of polynomial ideals, we will see that a Grobner basis can be computed from the input polynomial set, thus ....

....homogeneous polynomials, where it produces the exact resultant) For the Macaulay formulation, Canny [Can90] has invented a general method that perturbs any polynomial system and extracts a non trivial projection operator. Using recent results pertaining to sparse polynomial systems [GKZ94, Stu91, SZ92] the mixed sparse resultant of a system of n 1 sparse polynomials in n variables in its matrix form was given by Canny and Emiris [CE93a] and consequently improved in [CE93b, CE95] Here, sparsity denotes that only certain monomials in each of the n 1 polynomials have non zero ....

B. Sturmfels. Sparse elimination theory. In D. Eisenbud and L. Robbiano, editors, Proc. Computat. Algebraic Geom. and Commut. Algebra, Cortona, Italy, June 1991.

....input polynomials, but on their Newton polytopes. Besides the Dixon formulation, there are at least two other matrix based methods for computing multivariate projection operators Macaulay s formulation [11] developed early this century, and the recently developed sparse resultant formulation [13, 7]. Macaulay s formulation uses the traditional Bezout bound on the number of common solutions of a polynomial system to construct the Macaulay matrix, whose size depends on the total degree of the input polynomials. Hence Macaulay s formulation does not take the sparsity of input polynomials into ....

Sturmfels B., Sparse Elimination Theory, Proc. Computat. Algebraic Geom. and Commut. Algebra, D. Eisenbud and L. Robbiano, eds., Cortona , Italy, June 1991.

....Sederberg for implicitizing parametric curves and surfaces [6] The problem of curve intersection is reduced to solving two polynomial equations simultaneously and therefore we use resultant formulations of two polynomials in one unknown. Surveys on various formulations of resultants are given in [6, 27] and effective techniques for computing and applying them are presented in [14] Three methods are known in the literature for computing the resultant, owing to Bezout or Cayley and Sylvester [26] Each of them expresses the resultant as the determinant of a matrix. The order of the matrix is ....

B. Sturmfels. Sparse elimination theory. In D. Eisenbud and L. Robbiano, editors, Computational Algebraic Geometry and Commutative Algebra. Cambridge University Press, 1991.

.... similarly to Cayley s method [16] Lastly, our construction is closely related to that of Macauley s [13] More recently, the sparse unmixed resultant was defined as the Chow form of a projective toric variety in [10] see also [4] Algorithms for its computation and evaluation were proposed in [20], the most efficient one having complexity higher than polynomial in the degree of the resultant and exponential in n with a quadratic exponent. For multigraded systems, an optimal determinantal formula, called of Sylvester type, is given in [22] These systems are unmixed and include polynomials ....

Sturmfels, B.: Sparse elimination theory. In D. Eisenbud and L. Robbiano, editors, Proc. Computat. Algebraic Geom. and Commut. Algebra, Cortona, Italy, June 1991. Cambridge Univ. Press. To appear.

....theory are presented in [1, 49] Most of the formulation presented in the classical literature correspond to computing the resultants of dense polynomial systems. In the last few decades, a number of results have appeared in the literature pertaining to the resultants of sparse polynomial systems [2, 18, 44]. This is very useful due to the fact that many polynomial systems encountered in applications are rather sparse [34] Resultant formulations of sparse polynomial systems, expressed in terms of matrices and determinants, have appeared in [11, 45] 3. MULTIVARIATE INTERPOLATION The algorithms for ....

B. Sturmfels. Sparse elimination theory. In D. Eisenbud and L. Robbiano, editors, Computational Algebraic Geometry and Commutative Algebra. Cambridge University Press, 1991.

.... of fast algorithms for solving polynomial systems with finitely many solutions [18] resurrection and variations of classical constructive techniques for eliminating variables [5, 33] development of elimination methods which exploit the structure of polynomial systems to solve them efficiently [14, 7, 40], and the development of efficient techniques for numerically solving non linear systems [35, 42] Three major techniques for eliminating variables symbolically are: 1. Grobner basis computations proposed by Buchberger [4] 2. Characteristic set computations proposed by Ritt [37] and 3. Resultant ....

....zero projection operator. Also, under coefficient specializations, this formulation may give rise to extraneous factors. 2. 4 Sparse Resultants The Sparse resultant approach is based on the recent results pertaining to sparse polynomial systems (Bernstein [3] Gelfand et al. 23] and Sturmfels [40]) Sparse resultants first appeared in Sturmfels Zelevinsky [41] its matrix formulation was given by Canny Emiris [6] and was successively improved in [7, 15, 16] using better heuristics. The Macaulay formulation is based on the Bezout bound on the number of solutions to a set of ....

[Article contains additional citation context not shown here]

Sturmfels B., Sparse Elimination Theory, Proc. Computat. Algebraic Geom. and Commut. Algebra, D. Eisenbud and L. Robbiano, eds., Cortona, Italy, June 1991.

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B. Sturmfels. Sparse elimination theory. In Computational Algebraic Geometry and Commutative Algebra. Cambridge University Press, June 1991.

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B. Sturmfels. Sparse elimination theory. In D. Eisenbud and L. Robbiano, editors, Proc. Computat. Algebraic Geom. and Commut. Algebra 1991, pages 264--298, Cortona, Italy, 1993. Cambridge Univ. Press. 26

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Bernd Sturmfels. Sparse elimination theory. Sympos. Math., XXXIV, Cambridge Univ. Press, pages 264--298, 1993.

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B. Sturmfels. Sparse elimination theory. In D. Eisenbud and L. Robianno, editors, Computational Algebraic Geometry and Commutative Algebra, pages 264298. Cambridge Univ. Press, 1993.

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Sturmfels, B. (1993). Sparse elimination theory. In Eisenbud, D., Robianno, L., editors, Computational Algebraic Geometry and Commutative Algebra, pages 264--298. Cambridge Univ. Press. (Proc.

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Sturmfels, B.: Sparse elimination theory. In D. Eisenbud and L. Robbiano, editors, Proc. Computat. Algebraic Geom. and Commut. Algebra, Cortona, Italy, June 1991. Cambridge Univ. Press. To appear.

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B. Sturmfels. Sparse elimination theory. In D. Eisenbud and L. Robbiano, editors, Computational Algebraic Geometry and Commutative Algebra. Cambridge University Press, 1991.

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