Singular Zeros of Polynomial Systems

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

2016

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We bound the Boolean complexity of computing isolating hyperboxes for all complex roots of systems of bilinear polynomials. The resultant of such systems admits a family of determinantal Sylvester-type formulas, which we make explicit by means of homological complexes. The computation of the determinant of the resultant matrix is a bottleneck for the overall complexity. We exploit the quasi-Toeplitz structure to reduce the problem to efficient matrix-vector products, corresponding to multivariate polynomial multiplication. For zero-dimensional systems, we arrive at a primitive element and a rational univariate representation of the roots. The overall bit complexity of our probabilistic algorithm is OB(n 4 D 4 + n 2 D 4 τ ), where n is the number of variables, D equals the bilinear Bézout bound, and τ is the maximum coefficient bitsize. Finally, a careful infinitesimal symbolic perturbation of the system allows us to treat degenerate and positive dimensional systems, thus making our algorithms and complexity analysis applicable to the general case.√ 3 3 ; −1, 2 ∓ √ 3) plus the root (x0, x1; y0, y1, y2) = (0, 1; 0, −1, 1) at infinity.

Section: The Weyman Resultant Complexmentioning

confidence: 99%

On the Bit Complexity of Solving Bilinear Polynomial Systems

Emiris

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

2016

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“…, see [30,31] and references therein for a detailed presentation. We use this identification to obtain a basis for S * 2 (d).…”

Section: Primal-dual Multiplication Mapsmentioning

confidence: 99%

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

Busé

Journal of Symbolic Computation

2020

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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.

“…This space is isomorphic to (evaluations of) polynomials in formal derivatives, formally S * 2 (d) ∼ = R[∂y] d , see [37,38] and references therein for a detailed presentation. We use this identification to obtain a basis for S * 2 (d).…”

Section: Primal-dual Multiplication Mapsmentioning

confidence: 99%

Resultants and Discriminants for Bivariate Tensor-Product Polynomials

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

2017

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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.