An elementary approach to subresultants theory

Kahoui, M’hammed El

doi:10.1016/s0747-7171(02)00135-9

Cited by 28 publications

(27 citation statements)

References 14 publications

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“…For more details on subresultants theory we refer to [8,4,12,13,3,11,6], but the list is nowhere near exhaustive.…”

Section: Review Of Subresultantsmentioning

confidence: 99%

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2005

Proc. Amer. Math. Soc.

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“…For more details on subresultants theory we refer to [8,4,12,13,3,11,6], but the list is nowhere near exhaustive.…”

Section: Review Of Subresultantsmentioning

confidence: 99%

“…Let us point out that the results given in this section are elementary insofar as they are mainly due to the basic properties of polynomials and nice behavior of determinants under row and column operations (see e.g [6] for more details).…”

Section: Theorem 24 Let a Be A Ring And Let F G H ∈ A[t]mentioning

confidence: 99%

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2005

Proc. Amer. Math. Soc.

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“…Note that this property is often stated with a stronger assumption that is that none of the leading terms ap(α) and bq(α) vanishes. This property is a direct consequence of the specialization property of subresultants and of the gap structure theorem; see for instance [7 Lemma 2. For any α such that ap(α) and bq(α) do not both vanish, the first Sres Y,k (P, Q)(α, Y ) (for k increasing) that does not identically vanish is of degree k and it is the gcd of P (α, Y ) and Q(α, Y ) (up to a nonzero constant in the fraction field of D(α)).…”

Section: Notation and Preliminariesmentioning

confidence: 92%

Separating linear forms for bivariate systems

Bouzidi

Lazard

Pouget

et al. 2013

Proceedings of the 38th International Symposium on Symbolic and Algebraic Computation

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We present an algorithm for computing a separating linear form of a system of bivariate polynomials with integer coefficients, that is a linear combination of the variables that takes different values when evaluated at distinct (complex) solutions of the system. In other words, a separating linear form defines a shear of the coordinate system that sends the algebraic system in generic position, in the sense that no two distinct solutions are vertically aligned. The computation of such linear forms is at the core of most algorithms that solve algebraic systems by computing rational parameterizations of the solutions and, moreover, the computation of a separating linear form is the bottleneck of these algorithms, in terms of worst-case bit complexity.Given two bivariate polynomials of total degree at most d with integer coefficients of bitsize at most τ , our algorithm computes a separating linear form in e OB(d2 ) bit operations in the worst case, where the previously known best bit complexity for this problem was e OB(d(where e O refers to the complexity where polylogarithmic factors are omitted and OB refers to the bit complexity).

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“…Brown's algorithm on the GPU One of the most important algebraic tools for solving polynomial system symbolically is the polynomial subresultants. We refer to [7,21] for the theory of polynomial subresultants. In this section, after briefly reviewing some properties of polynomial subresultants, we present our CUDA FFT-based implementation of Brown's subresultant algorithm for bivariate (or trivariate) polynomials over a prime field K = Z c .…”

Section: Formentioning

confidence: 99%

Solving Bivariate Polynomial Systems on a GPU

Maza

Pan²

2011

AIP Conference Proceedings

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Abstract. We present a CUDA implementation of dense multivariate polynomial arithmetic based on Fast Fourier Transforms over finite fields. Our core routine computes on the device (GPU) the subresultant chain of two polynomials with respect to a given variable. This subresultant chain is encoded by values on a FFT grid and is manipulated from the host (CPU) in higher-level procedures.We have realized a bivariate polynomial system solver supported by our GPU code. Our experimental results (including detailed profiling information and benchmarks against a serial polynomial system solver implementing the same algorithm) demonstrate that our strategy is well suited for GPU implementation and provides large speedup factors with respect to pure CPU code.

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An elementary approach to subresultants theory

Cited by 28 publications

References 14 publications

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Separating linear forms for bivariate systems

Solving Bivariate Polynomial Systems on a GPU

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