Obtaining Exact Interpolation Multivariate Polynomial by Approximation

Sparse Interpolation in Terms of Multivariate Chebyshev Polynomials

Sparse interpolation} refers to the exact recovery of a function as a short linear combination of basis functions from a limited number of evaluations. For multivariate functions, the case of the monomial basis is well studied, as is now the basis of exponential functions. Beyond the multivariate Chebyshev polynomial obtained as tensor products of univariate Chebyshev polynomials, the theory of root systems allows to define a variety of generalized multivariate Chebyshev polynomials that have connections to topics such as Fourier analysis and representations of Lie algebras. We present a deter…

Sparse Polynomial Interpolation with Finitely Many Values for the Coefficients

In this paper, we give new sparse interpolation algorithms for black box polynomial f whose coefficients are from a finite set. In the univariate case, we recover f from one evaluation of f(a) for a sufficiently large number a. In the multivariate case, we introduce the modified Kronecker substitution to reduce the interpolation of a multivariate polynomial to the univariate case. Both algorithms have polynomial bit-size complexity.

Sparse Rational Function Interpolation with Finitely Many Values for the Coefficients

In this paper, we give new sparse interpolation algorithms for black box univariate and multivariate rational functions h=f/g whose coefficients are integers with an upper bound. The main idea is as follows: choose a proper integer beta and let h(beta) = a/b with gcd(a,b)=1. Then f and g can be computed by solving the polynomial interpolation problems f(beta)=ka and g(beta)=ka for some integer k. It is shown that the univariate interpolation algorithm is almost optimal and multivariate interpolation algorithm has low complexity in T but the data size is exponential in n.

Faster Algorithms for Multivariate Interpolation with Multiplicities and Simultaneous Polynomial Approximations

The interpolation step in the Guruswami-Sudan algorithm is a bivariate interpolation problem with multiplicities commonly solved in the literature using either structured linear algebra or basis reduction of polynomial lattices. This problem has been extended to three or more variables; for this generalization, all fast algorithms proposed so far rely on the lattice approach. In this paper, we reduce this multivariate interpolation problem to a problem of simultaneous polynomial approximations, which we solve using fast structured linear algebra. This improves the best known complexity bounds …

Diversification improves interpolation

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 evaluations. Using related techniques, we also address numerical interpolation of sparse polynomials with complex coefficients, and provide the first provably stable algorithm (in the sense of relative error) for this problem,…

多元多项式的标准和三角行列式表示

Normal and Triangular Determinantal Representations of Multivariate Polynomials

本文提出了一种新的简单算法，用于将任意多元多项式转化为标准行列式形式，其中每个条目具有特定形式，且每列中的相同变量出现。该算法还用于获得多元多项式的三角行列式表示、简化的行列式表示和统一的行列式表示。该算法可能有助于获得比目前可用的更小维度的表示以解决数值问题。

Sparse Polynomial Interpolation Based on Derivative

In this paper, we propose two new interpolation algorithms for sparse multivariate polynomials represented by a straight-line program(SLP). Both of our algorithms work over any finite fields $F_q$ with large characteristic. The first one is a Monte Carlo randomized algorithm. Its arithmetic complexity is linear in the number $T$ of non-zero terms of $f$, in the number $n$ of variables. If $q$ is $O((nTD)^{(1)})$, where $D$ is the partial degree bound, then our algorithm has better complexity than other existing algorithms. The second one is a deterministic algorithm. It has better complexity t…

Computing minimal interpolation bases

We consider the problem of computing univariate polynomial matrices over a field that represent minimal solution bases for a general interpolation problem, some forms of which are the vector M-Padé approximation problem in [Van Barel and Bultheel, Numerical Algorithms 3, 1992] and the rational interpolation problem in [Beckermann and Labahn, SIAM J. Matrix Anal. Appl. 22, 2000]. Particular instances of this problem include the bivariate interpolation steps of Guruswami-Sudan hard-decision and Kötter-Vardy soft-decision decodings of Reed-Solomon codes, the multivariate interpolation step of lis…