On sparse interpolation of rational functions and gcds

Hoeven, Joris van der; Lecerf, Grégoire

doi:10.1145/3466895.3466896

Cited by 5 publications

(2 citation statements)

References 19 publications

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“…is algorithm is given in the context of an unknown polynomial that a black box allows to evaluate at any point of ℤ freely chosen by the algorithm. Numerous extensions have been proposed [51,38,30], in particular in order to: deal with finite fields [21,26,32,16,29], avoid the bound on by early termination techniques [34] or extend the problem to the case of sparse rational function [39,37,12,25]. Some algorithms require the black box model to be slightly relaxed and allow evaluations in extension rings or quotient rings [21,41,2,45,37,16,10,23].…”

Section: Introductionmentioning

confidence: 99%

Sparse Polynomial Interpolation and Division in Soft-linear Time

Giorgi,

Grenet,

Cray

et al. 2022

Preprint

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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. ese 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 an integer polynomial ( ) given a way to evaluate ( ) mod for any chosen integers and . is algorithm has nearly-optimal bit complexity, meaning that the total bit-length of the probes, as well as the computational running time, is so ly linear (ignoring logarithmic factors) in the bit-length of the resulting sparse polynomial. To our knowledge, this is the first sparse interpolation algorithm with so -linear bit complexity in the total output size. For integer polynomials, the best previously known results have at least a cubic dependency on the bit-length of the exponents. * In this work, we do not consider the case of unbalanced bit lengths, where the differing sizes of each coefficient and exponent are considered in the complexity.

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Section: Introductionmentioning

confidence: 99%

Sparse Polynomial Interpolation and Division in Soft-linear Time

Giorgi,

Grenet,

Cray

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…This algorithm is given in the context of an unknown polynomial that a black box allows to evaluate at any point of Z freely chosen by the algorithm. Numerous extensions have been proposed [30,38,52], in particular in order to: deal with finite fields [16,21,26,29,32], avoid the bound on 𝑡 by early termination techniques [34] or extend the problem to the case of sparse rational functions [12,25,37,39]. Some algorithms require the black box model to be slightly relaxed and allow evaluations in extension rings or quotient rings [2,10,16,21,23,37,41,45].…”

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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