Parallel sparse multivariate polynomial division

Laskar, Jacques

doi:10.1145/2790282.2790285

Cited by 9 publications

(5 citation statements)

References 19 publications

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“…Formally analyzing algorithms (in the ideal-cache model) which perform arithmetic operations on sparse polynomials is work in progress. Nevertheless, practically efficient algorithms for multiplying and dividing sparse polynomials are available and implemented, see Monagan and Pearce [44,45], Gastineau and Laskard [29,30], the MSc thesis of Brandt [12] for a detailed account, as well as [3] from the BPAS developers.…”

Section: Dense Univariate Polynomial Arithmetic Over a Finite Fieldmentioning

confidence: 99%

Design and Implementation of Multi-Threaded Algorithms in Polynomial Algebra

Maza

2021

Proceedings of the 2021 International Symposium on Symbolic and Algebraic Computation

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Section: Dense Univariate Polynomial Arithmetic Over a Finite Fieldmentioning

confidence: 99%

Design and Implementation of Multi-Threaded Algorithms in Polynomial Algebra

Maza

2021

Proceedings of the 2021 International Symposium on Symbolic and Algebraic Computation

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“…When dividing sparse polynomials, it is imperative to consider the output size: for example, the exact division of two 2-term polynomials x D − 1 by x − 1 produces a quotient with D nonzero terms. Fortunately, the heaps idea which works well in practice for sparse multiplication has also been adapted to sparse division, and this method easily yields the remainder as well as the quotient [31,71]. As before, this approach uses O t 2 ring operations, leaving us with another challenge:…”

Section: Divisionmentioning

confidence: 99%

What Can (and Can't) we Do with Sparse Polynomials?

Roche

2018

Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation

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Simply put, a sparse polynomial is one whose zero coefficients are not explicitly stored. Such objects are ubiquitous in exact computing, and so naturally we would like to have efficient algorithms to handle them. However, with this compact storage comes new algorithmic challenges, as fast algorithms for dense polynomials may no longer be efficient. In this tutorial we examine the state of the art for sparse polynomial algorithms in three areas: arithmetic, interpolation, and factorization. The aim is to highlight recent progress both in theory and in practice, as well as opportunities for future work. KEYWORDSsparse polynomial, interpolation, arithmetic, factorization ACM Reference Format: Daniel S. Roche. 2018. What Can (and Can't) we Do with Sparse Polynomials?.

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“…Even for standard operations such as multiplication or division, no deterministic quasi-linear time algorithm is known. In spite of some theoretical improvements and practical implementations, deterministic algorithms for these operations remain quadratic in the sparsity [15,33,[42][43][44]. The major difficulty comes from the unpredictability of the sparsity of the result.…”

Section: Introductionmentioning

confidence: 99%

Sparse Polynomial Interpolation and Division in Soft-linear Time

Giorgi

Grenet

Cray

et al. 2022

Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation

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Given a way to evaluate an unknown polynomial with integer coefficients, we present new algorithms to recover its nonzero coefficients and corresponding exponents. As an application, we adapt this interpolation algorithm to the problem of computing the exact quotient of two given polynomials. These methods are efficient in terms of the bit-length of the sparse representation, that is, the number of nonzero terms, the size of coefficients, the number of variables, and the logarithm of the degree. At the core of our results is a new Monte Carlo randomized algorithm to recover a polynomial 𝑓 (𝑥) with integer coefficients given a way to evaluate 𝑓 (𝜃 ) mod 𝑚 for any chosen integers 𝜃 and 𝑚. This algorithm has nearly-optimal bit complexity, meaning that the total bit-length of the probes, as well as the computational running time, is softly linear (ignoring logarithmic factors) in the bit-length of the resulting sparse polynomial. To our knowledge, this is the first sparse interpolation algorithm with soft-linear bit complexity in the total output size. For polynomials with integer coefficients, the best previously known results have at least a cubic dependency on the bit-length of the exponents. CCS CONCEPTS• Computing methodologies → Algebraic algorithms; • Theory of computation → Design and analysis of algorithms; • Mathematics of computing → Probabilistic algorithms.

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Parallel sparse multivariate polynomial division

Cited by 9 publications

References 19 publications

Design and Implementation of Multi-Threaded Algorithms in Polynomial Algebra

Design and Implementation of Multi-Threaded Algorithms in Polynomial Algebra

What Can (and Can't) we Do with Sparse Polynomials?

Sparse Polynomial Interpolation and Division in Soft-linear Time

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