Conditions for exact resultants using the Dixon formulation

Chtcherba, Arthur D.; Kapur, Deepak

doi:10.1145/345542.345583

Cited by 15 publications

(16 citation statements)

References 9 publications

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“…The third contribution of this paper is to offer sufficient and necessary conditions for systems to admit purely Bézout bilateral collaboration and FET Open European Project IST-2001-35512 (GAIA-II). determinantal formulae, thus generalizing a result from [CK00]. It turns out that these are precisely the same systems admitting optimal Sylvester-type formulae, and this is nothing but a special case of complexes with only two non-vanishing cohomologies.…”

Section: Introductionmentioning

confidence: 52%

Multihomogeneous resultant formulae by means of complexes

Dickenstein

Emiris

2003

Journal of Symbolic Computation

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The first step in the generalization of the classical theory of homogeneous equations to the case of arbitrary support is to consider algebraic systems with multihomogeneous structure. We propose constructive methods for resultant matrices in the entire spectrum of resultant formulae, ranging from pure Sylvester to pure Bézout types, and including matrices of hybrid type of these two. Our approach makes heavy use of the combinatorics of multihomogeneous systems, inspired by and generalizing certain joint results by Zelevinsky, and Sturmfels or Weyman [SZ94], [WZ94]. One contribution is to provide conditions and algorithmic tools so as to classify and construct the smallest possible determinantal formulae for multihomogeneous resultants. Whenever such formulae exist, we specify the underlying complexes so as to make the resultant matrix explicit. We also examine the smallest Sylvester-type matrices, generically of full rank, which yield a multiple of the resultant. The last contribution is to characterize the systems that admit a purely Bézout-type matrix and show a bijection of such matrices with the permutations of the variable groups. Interestingly, it is the same class of systems admitting an optimal Sylvester-type formula. We conclude with examples showing the kinds of matrices that may be encountered, and illustrations of our Maple implementation.

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Section: Introductionmentioning

confidence: 52%

Multihomogeneous resultant formulae by means of complexes

Dickenstein

Emiris

2003

Journal of Symbolic Computation

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“…While they establish that any maximal minor of the Dixon coefficient matrix is a multiple of the A-resultant, this paper gives a sufficient condition for a unique maximal minor which is the A-resultant so there are no extraneous factors. Other results on the persistence of Dixon formulation and rectangular corner cutting can be found in Chtcherba and Kapur (2000) and Zhang and Goldman (2000).…”

Section: Introductionmentioning

confidence: 90%

Rectangular Corner Cutting and Dixon A -resultants

Chionh

2001

Journal of Symbolic Computation

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The classical Dixon resultant for three bi-degree polynomials in two variables is an A-resultant with a rectangular bi-degree monomial support. This paper shows that the determinant of the Dixon coefficient matrix persists as an A-resultant after rectangular corner cutting, that is, the removal of one or more rectangular sub-supports at the corners of the bi-degree support. The proof is constructive and produces intermediate results which are significant in their own right.

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“…The degree of ∆ d 1 ,d 2 (f ) follows directly by looking at the degree of the denominator and numerator of (16) .…”

Section: Discriminants Of Tp Polynomialsmentioning

confidence: 99%

“…His determinantal formula is a hybrid Sylvester and Bézout type. Also in the unmixed case there are necessary and sufficient conditions for the Dixon resultant formulation to produce the resultant [15,16]. In the same context, Elkadi and Galligo proposed in [23] to use a variable substitution and two iterated resultants to compute the resultant polynomial.…”

Section: Introductionmentioning

confidence: 99%

Resultants and Discriminants for Bivariate Tensor-Product Polynomials

Mantzaflaris

Tsigaridas

2017

Proceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation

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Optimal resultant formulas have been systematically constructed mostly for unmixed polynomial systems, that is, systems of polynomials which all have the same support. However, such a condition is restrictive, since mixed systems of equations arise frequently in practical problems. We present a square, Koszul-type matrix expressing the resultant of arbitrary (mixed) bivariate tensor-product systems. The formula generalizes the classical Sylvester matrix of two univariate polynomials, since it expresses a map of degree one, that is, the entries of the matrix are simply coefficients of the input polynomials. Interestingly, the matrix expresses a primaldual multiplication map, that is, the tensor product of a univariate multiplication map with a map expressing derivation in a dual space. Moreover, for tensor-product systems with more than two (affine) variables, we prove an impossibility result: no universal degree-one formulas are possible, unless the system is unmixed. We present applications of the new construction in the computation of discriminants and mixed discriminants as well as in solving systems of bivariate polynomials with tensor-product structure.

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Conditions for exact resultants using the Dixon formulation

Cited by 15 publications

References 9 publications

Multihomogeneous resultant formulae by means of complexes

Multihomogeneous resultant formulae by means of complexes

Rectangular Corner Cutting and Dixon A -resultants

Resultants and Discriminants for Bivariate Tensor-Product Polynomials

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