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Fast Parallel Algorithms for Sparse Multivariate Polynomial Interpolation over Finite Fields

Dima Yu. Grigoriev, Marek Karpinski, Michael F. Singer
1990 SIAM journal on computing (Print)  
Key words, 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.  ... 
doi:10.1137/0219073 fatcat:ms2ohrjgqjddjprlwezcjesqn4

Diversification improves interpolation [article]

Mark Giesbrecht, Daniel S. Roche
2011 arXiv   pre-print
We consider the problem 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.  ... 
arXiv:1101.3682v3 fatcat:zavsjkp4cjaafd77hgav7dmwce

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

Mark Giesbrecht, Daniel S. Roche
2011 Proceedings of the 36th international symposium on Symbolic and algebraic computation - ISSAC '11  
Background 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.  ... 
doi:10.1145/1993886.1993909 dblp:conf/issac/GiesbrechtR11 fatcat:7rmfxz4iwfaeto7vwpfxhc4bcq

Special issue computational algebraic complexity editorial

Erich Kaltofen, Bruno Buchberger
1990 Journal of symbolic computation  
The article by Zippel deals 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.  ... 
doi:10.1016/s0747-7171(08)80010-7 fatcat:wrri7sabmjb4tpqtqyfbwuoa6e

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]

Qiao-Long Huang
2020 arXiv   pre-print
We consider the problem 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.  ... 
arXiv:2002.03706v1 fatcat:2dngbk4z3bdktihacrjma3tm4e

Reconstructing Rational Functions with FireFly [article]

Jonas Klappert, Fabian Lange
2019 arXiv   pre-print
We present the open-source C++ library FireFly for the reconstruction 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] .  ... 
arXiv:1904.00009v1 fatcat:ksgwhettkrfodejxpp4epvf7ge

Model Reduction Using Sparse Polynomial Interpolation for the Incompressible Navier-Stokes Equations [article]

Martin W. Hess, Gianluigi Rozza
2022 arXiv   pre-print
Numerical results are presented underscoring the validity 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  ... 
arXiv:2201.03228v1 fatcat:u2brlztnnfafhdezbthht5vkxa

Efficient techniques for multipolynomial resultant algorithms

Dinesh Manocha, John Canny
1991 Proceedings of the 1991 international symposium on Symbolic and algebraic computation - ISSAC '91  
upon the 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.  ... 
doi:10.1145/120694.120706 dblp:conf/issac/ManochaC91 fatcat:moothyqmyrbmbpuuyqow5azm5y

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  ... 
doi:10.1201/b15006-5 fatcat:cubpnr7y3fbfpivjinvw2dqmvy

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  ... 
doi:10.1201/b10783-7 fatcat:zqyyjtuzsjf7zpekrozupsqwye

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  ... 
doi:10.1142/9789814596077_0009 fatcat:7rmqxc74afagha2moms4jjz35u

A deterministic multivariate interpolation algorithm for small finite fields

Z. Zilic, Z.G. Vranesic
2002 IEEE transactions on computers  
We present a new 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.  ... 
doi:10.1109/tc.2002.1032628 fatcat:3kzrygdxrffc5fuf2rfg54aoky

Sparse interpolation of multivariate rational functions

Annie Cuyt, Wen-shin Lee
2011 Theoretical Computer Science  
Our method can be combined 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.  ... 
doi:10.1016/j.tcs.2010.11.050 fatcat:pmnhbkkj3zethawvfmovzdnohq
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