Resultants for unmixed bivariate polynomial systems produced using the Dixon formulation

Chtcherba, Arthur D.; Kapur, Deepak

doi:10.1016/j.jsc.2003.12.001

Cited by 11 publications

(13 citation statements)

References 9 publications

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“…Finally using (15) and (16), (23) can be changed into (18). Now suppose that p u1,u2,...,un = n+1 j=1 (−1) j+1 k j,u1,u2,...,un f j , where k j,u1,u2,...,un is a polynomial with respect to {x 1 , .…”

Section: Three Kinds Of Extraneous Factors In Dixon Resultantsmentioning

confidence: 99%

Three kinds of extraneous factors in Dixon resultants

Zhao

2008

Sci. China Ser. A-Math.

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Dixon resultant is a basic elimination method which has been used widely in the high technology fields of automatic control, robotics, etc. But how to remove extraneous factors in Dixon resultants has been a very difficult problem. In this paper, we discover some extraneous factors by expressing the Dixon resultant in a linear combination of original polynomial system. Furthermore, it has been proved that the factors mentioned above include three parts which come from Dixon derived polynomials, Dixon matrix and the resulting resultant expression by substituting Dixon derived polynomials respectively.

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Section: Three Kinds Of Extraneous Factors In Dixon Resultantsmentioning

confidence: 99%

Three kinds of extraneous factors in Dixon resultants

Zhao

2008

Sci. China Ser. A-Math.

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“…The result follows from [32, Chapter 10,Th. 1.2], which we specialize for f (x, y) in (15). That theorem states that the discriminant…”

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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“…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 [7,8]. In the same context, Elkadi and Galligo proposed in [17] to use a variable substitution and two iterated resultants to compute the resultant polynomial.…”

Section: Introductionmentioning

confidence: 99%

Matrix formulæ for resultants and discriminants of bivariate tensor-product polynomials

Busé

Mantzaflaris

Tsigaridas

2020

Journal of Symbolic Computation

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The construction of optimal resultant formulae for polynomial systems is one of the main areas of research in computational algebraic geometry. However, most of the constructions are restricted to formulae for unmixed polynomial systems, that is, systems of polynomials which all have the same support. Such a condition is restrictive, since mixed systems of equations arise frequently in many problems. Nevertheless, resultant formulae for mixed polynomial systems is a very challenging problem. We present a square, Koszul-type, matrix, the determinant of which is the resultant of an arbitrary (mixed) bivariate tensor-product polynomial system. The formula generalizes the classical Sylvester matrix of two univariate polynomials, since it expresses a map of degree one, that is, the elements of the corresponding matrix are up to sign the coefficients of the input polynomials. Interestingly, the matrix expresses a primal-dual multiplication map, that is, the tensor product of a univariate multiplication map with a map expressing derivation in a dual space. In addition we prove an impossibility result which states that for tensor-product systems with more than two (affine) variables there are no universal degree-one formulae, unless the system is unmixed. Last but not least, we present applications of the new construction in the efficient computation of discriminants and mixed discriminants.

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Resultants for unmixed bivariate polynomial systems produced using the Dixon formulation

Cited by 11 publications

References 9 publications

Three kinds of extraneous factors in Dixon resultants

Three kinds of extraneous factors in Dixon resultants

Resultants and Discriminants for Bivariate Tensor-Product Polynomials

Matrix formulæ for resultants and discriminants of bivariate tensor-product polynomials

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