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Fast Parallel Algorithms for Sparse Multivariate Polynomial Interpolation over Finite Fields

1990
*
SIAM journal on computing (Print)
*

Key words,

doi:10.1137/0219073
fatcat:ms2ohrjgqjddjprlwezcjesqn4
*sparse**multivariate**polynomials*,*finite**fields*,*interpolation*AMS(MOS) subject classifications. 68C25, 12C05 GF[qr21g"n'+3], and (2) using inductive enumeration*of*partial solutions for terms ... This algorithm yields the first efficient deterministic*polynomial*time algorithm (and moreover boolean NC-algorithm) for*interpolating*t-*sparse**polynomials**over**finite**fields*and should be contrasted ... We are grateful to Michael Ben-Or, Johannes Grabmeier, Michael Rabin, Volker Strassen, and Avi Wigderson for a number*of*interesting conversations. ...##
###
Diversification improves interpolation
[article]

2011
*
arXiv
*
pre-print

We consider the problem

arXiv:1101.3682v3
fatcat:zavsjkp4cjaafd77hgav7dmwce
*of**interpolating*an unknown*multivariate**polynomial**with*coefficients taken from a*finite**field*or as numerical approximations*of*complex numbers. ... Building on the recent work*of*Garg and Schost, we improve on the best-known algorithm for*interpolation**over**large**finite**fields*by presenting a Las Vegas randomized algorithm that uses fewer black box ... The comments and suggestions*of*the anonymous referees were also very helpful, in particular regarding connections to previous results and the proof*of*Theorem 3.1. ...##
###
Page 834 of Mathematical Reviews Vol. , Issue 97B
[page]

1997
*
Mathematical Reviews
*

Summary: “We develop a randomized parallel algorithm which performs

*interpolation**of**sparse**multivariate**polynomials**over**finite**fields*. ... As*applications*, we obtain efficient parallel algorithms for*sparse**multivariate**polynomial*factorization and GCD*over**finite**fields*. ...##
###
Diversification improves interpolation

2011
*
Proceedings of the 36th international symposium on Symbolic and algebraic computation - ISSAC '11
*

Background

doi:10.1145/1993886.1993909
dblp:conf/issac/GiesbrechtR11
fatcat:7rmfxz4iwfaeto7vwpfxhc4bcq
*Finite**Fields*Approximate Complex*Multivariate*and beyond*Sparse**interpolation*algorithms*over**finite**fields*• Dense methods (Newton/Waring/Lagrange): O˜(d) total cost. ... We first consider univariate*interpolation**over**finite**fields*. ...##
###
Special issue computational algebraic complexity editorial

1990
*
Journal of symbolic computation
*

The article by Zippel deals

doi:10.1016/s0747-7171(08)80010-7
fatcat:wrri7sabmjb4tpqtqyfbwuoa6e
*with*the problem*of**interpolating*a*sparse**multivariate**polynomial**over**fields**of*characteristic zero from its values. ... result in a collection*of*surprisingly effective algorithms for*sparse**multivariate**polynomial**interpolation*, which are presented here. ...##
###
Page 815 of Mathematical Reviews Vol. , Issue 2001B
[page]

2001
*
Mathematical Reviews
*

*of*

*sparse*

*multivariate*

*polynomials*

*over*

*large*

*finite*

*fields*

*with*

*applications*. ... The authors develop a Las Vegas randomized algorithm for

*sparse*

*polynomial*

*interpolation*

*over*

*finite*

*fields*. ...

##
###
Sparse Polynomial Interpolation Based on Diversification
[article]

2020
*
arXiv
*
pre-print

We consider the problem

arXiv:2002.03706v1
fatcat:2dngbk4z3bdktihacrjma3tm4e
*of**interpolating*a*sparse**multivariate**polynomial**over*a*finite**field*, represented*with*a black box. ... Building on the algorithm*of*Ben-Or and Tiwari for*interpolating**polynomials**over*rings*with*characteristic zero, we develop a new Monte Carlo algorithm*over*the*finite**field*by doing additional probes ... In 2001, Klivans and Spielman gave the first deterministic*polynomial*time algorithms for*sparse**interpolation**over**finite**fields**with**large*characteristic. ...##
###
Reconstructing Rational Functions with FireFly
[article]

2019
*
arXiv
*
pre-print

We present the open-source C++ library FireFly for the reconstruction

arXiv:1904.00009v1
fatcat:ksgwhettkrfodejxpp4epvf7ge
*of**multivariate*rational functions*over**finite**fields*. We discuss the involved algorithms and their implementation. ... As an*application*, we use FireFly in the context*of*integration-by-parts reductions and compare runtime and memory consumption to a fully algebraic approach*with*the program Kira. ... The Feynman diagrams in this paper were drawn*with*Tik Z-Feynman [75] . ...##
###
Model Reduction Using Sparse Polynomial Interpolation for the Incompressible Navier-Stokes Equations
[article]

2022
*
arXiv
*
pre-print

Numerical results are presented underscoring the validity

arXiv:2201.03228v1
fatcat:u2brlztnnfafhdezbthht5vkxa
*of**sparse**polynomial*approximations and comparing*with*established reduced basis techniques. ... The findings establish*sparse**polynomial**interpolation*as another instrument in the toolbox*of*methods for breaking the curse*of*dimensionality. ... Acknowledgements We acknowledge the support provided by the European Research Council Executive Agency by the Consolidator Grant project AROMA-CFD "Advanced Reduced Order Methods*with**Applications*in Computational ...##
###
Efficient techniques for multipolynomial resultant algorithms

1991
*
Proceedings of the 1991 international symposium on Symbolic and algebraic computation - ISSAC '91
*

upon the

doi:10.1145/120694.120706
dblp:conf/issac/ManochaC91
fatcat:moothyqmyrbmbpuuyqow5azm5y
*application*). 3.1*Multivariate**Interpolation*oft he resulting matrix is a*polynomial*. ... The timings correspond to a single iteration*over*a*finite**field*and typically 3 -4 iterations are required. ...##
###
Theoretical Properties
[chapter]

2013
*
Handbook of Finite Fields
*

*large*

*finite*

*field*are NP-or co-NP-hard. ... Here we have a situation where factoring

*over*the rational numbers is provably easier than factoring

*over*a sufficiently

*large*

*finite*

*field*. ... Minkowski sum, 9 Newton polytope, 9 NP-hardness, 11 Ostrowski theorem, 9 separable factorization, 3

*sparse*factorization, 9

*sparse*

*polynomial*representation, 9 straight-line program, 11, 12 straight-line ...

##
###
Theoretical Properties
[chapter]

2011
*
Chapman & Hall/CRC Biostatistics Series
*

*large*

*finite*

*field*are NP-or co-NP-hard. ... Here we have a situation where factoring

*over*the rational numbers is provably easier than factoring

*over*a sufficiently

*large*

*finite*

*field*. ... Minkowski sum, 9 Newton polytope, 9 NP-hardness, 11 Ostrowski theorem, 9 separable factorization, 3

*sparse*factorization, 9

*sparse*

*polynomial*representation, 9 straight-line program, 11, 12 straight-line ...

##
###
Theoretical Properties
[chapter]

2015
*
The Fence Methods
*

*large*

*finite*

*field*are NP-or co-NP-hard. ... Here we have a situation where factoring

*over*the rational numbers is provably easier than factoring

*over*a sufficiently

*large*

*finite*

*field*. ... Minkowski sum, 9 Newton polytope, 9 NP-hardness, 11 Ostrowski theorem, 9 separable factorization, 3

*sparse*factorization, 9

*sparse*

*polynomial*representation, 9 straight-line program, 11, 12 straight-line ...

##
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A deterministic multivariate interpolation algorithm for small finite fields

2002
*
IEEE transactions on computers
*

We present a new

doi:10.1109/tc.2002.1032628
fatcat:3kzrygdxrffc5fuf2rfg54aoky
*multivariate**interpolation*algorithm*over*arbitrary*fields*which is primarily suited for small*finite**fields*. ... Given function values at arbitrary t points, we show that it is possible to find an n-variable*interpolating**polynomial**with*at most t terms, using the number*of**field*operations that is*polynomial*in ... Anonymous reviewers provided invaluable help in improving the final presentation*of*the paper. ...##
###
Sparse interpolation of multivariate rational functions

2011
*
Theoretical Computer Science
*

Our method can be combined

doi:10.1016/j.tcs.2010.11.050
fatcat:pmnhbkkj3zethawvfmovzdnohq
*with*probabilistic and deterministic components from*sparse**polynomial*black box*interpolation*to suit either an exact or a*finite*precision computational environment. ... The latter is illustrated*with*several examples, running from exact*finite**field*arithmetic to noisy floating point evaluations. ... Acknowledgements We thank Erich Kaltofen and Zhengfeng Yang for valuable remarks and providing their*sparse*rational*interpolation*codes and benchmarks. ...
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