Improved algorithm for computing separating linear forms for bivariate systems

In this paper, we give improved bounds for the computational complexity of computing with planar algebraic curves. More specifically, for arbitrary coprime polynomials f , g ∈ Z[x, y] and an arbitrary polynomial h ∈ Z[x, y], each of total degree less than n and with integer coefficients of absolute value less than 2 τ , we show that each of the following problems can be solved in a deterministic way with a number of bit operations bounded byÕ(n 6 + n 5 τ), where we ignore polylogarithmic factors in n and τ:• The computation of isolating regions in C 2 for all complex solutions of the system f = g = 0,• the computation of a separating form for the solutions of f = g = 0,• the computation of the sign of h at all real valued solutions of f = g = 0, and• the computation of the topology of the planar algebraic curve C defined as the real valued vanishing set of the polynomial f .Our bound improves upon the best currently known bounds for the first three problems by a factor of n 2 or more and closes the gap to the state-of-the-art randomized complexity for the last problem.

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“…For the sum related to the lower bounds, the considerations in [24, Sec. 4.3.1] carry over 10 and yield…”

Section: Validation Phasementioning

confidence: 99%

“…Lemma 16. Let LB(α) be defined as in (10). Then, ∑ α log(LB(α) −1 ) = Õ(N 2 + NT), where we sum over all distinct complex roots of the resultant polynomial R (y) .…”

Section: A Missing Proofsmentioning

confidence: 99%

On the complexity of computing with planar algebraic curves

Kobel

Sagraloff

2015

Journal of Complexity

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“…In the univariate case the resultant variety is determinantal; it is given by the classical Sylvester matrix of the form (6). We have just shown that, in the bivariate tensorproduct case, there exists a formula, which does not have the structure of (6).…”

Section: The Determinantal Koszul Formulamentioning

confidence: 96%

“…We have just shown that, in the bivariate tensorproduct case, there exists a formula, which does not have the structure of (6). A natural question is whether any of these two (or any other) degree-one formulas are possible, unconditionally, for multivariate tensor-product systems.…”

Section: The Determinantal Koszul Formulamentioning

confidence: 98%

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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“…Computing the resultant of two polynomials is an ubiquitous question in symbolic computation, with applications to polynomial system solving [17], computational topology [18,2,12,7,8,5,22,6], Galois theory [28,24,1,23], computations with algebraic numbers [4], etc.…”

Section: Introductionmentioning

confidence: 99%

A Fast Algorithm for Computing the Truncated Resultant

Moroz

Schost

2016

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

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Let P and Q be two polynomials in K[x, y] with degree at most d, where K is a field. Denoting by R ∈ K[x] the resultant of P and Q with respect to y, we present an algorithm to compute R mod x k in O˜(kd) arithmetic operations in K, where the O˜notation indicates that we omit polylogarithmic factors. This is an improvement over state-of-the-art algorithms that require to compute R in O˜(d 3 ) operations before computing its first k coefficients.

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Improved algorithm for computing separating linear forms for bivariate systems

Cited by 6 publications

References 15 publications

On the complexity of computing with planar algebraic curves

On the complexity of computing with planar algebraic curves

Resultants and Discriminants for Bivariate Tensor-Product Polynomials

A Fast Algorithm for Computing the Truncated Resultant

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