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 than existing deterministic algorithms over a field with large characteristic. Its arithmetic complexity is quadratic in $n,T,\log D$, i.e., quadratic in the size of the sparse representation. And we also show that the complexity of our deterministic algorithm is the same as the one of deterministic zero-testing of Bläser et al. for the polynomial given by an SLP over finite field (for large characteristic).
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.
In this paper, a new reduction based interpolation algorithm for black-box multivariate polynomials over finite fields is given. The method is based on two main ingredients. A new Monte Carlo method is given to reduce black-box multivariate polynomial interpolation to black-box univariate polynomial interpolation over any ring. The reduction algorithm leads to multivariate interpolation algorithms with better or the same complexities most cases when combining with various univariate interpolation algorithms. We also propose a modified univariate Ben-or and Tiwarri algorithm over the finite fie…
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.
In this paper, we propose new deterministic and Monte Carlo interpolation algorithms for sparse multivariate polynomials represented by straight-line programs. Let $f$ be an $n$-variate polynomial given by a straight-line program, which has a degree bound $D$ and a term bound $T$. Our deterministic algorithm is quadratic in $n,T$ and cubic in $\log D$ in the Soft-Oh sense, which has better complexities than existing deterministic interpolation algorithms in most cases. Our Monte Carlo interpolation algorithms have better complexities than existing Monte Carlo interpolation algorithms and are t…
In this paper, we propose two new deterministic interpolation algorithms for a sparse multivariate polynomial given as a standard black-box by introducing new Kronecker type substitutions. Let $f\in \RB[x_1,\dots,x_n]$ be a sparse black-box polynomial with a degree bound $D$. When $\RB=\C$ or a finite field, our algorithms either have better bit complexity or better bit complexity in $D$ than existing deterministic algorithms. In particular, in the case of deterministic algorithms for standard black-box models, our second algorithm has the current best complexity in $D$ which is the dominant f…
We present a new Monte Carlo algorithm for the interpolation of a straight-line program as a sparse polynomial $f$ over an arbitrary finite field of size $q$. We assume a priori bounds $D$ and $T$ are given on the degree and number of terms of $f$. The approach presented in this paper is a hybrid of the diversified and recursive interpolation algorithms, the two previous fastest known probabilistic methods for this problem. By making effective use of the information contained in the coefficients themselves, this new algorithm improves on the bit complexity of previous methods by a "soft-Oh" fa…
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…
To interpolate a supersparse polynomial with integer coefficients, two alternative approaches are the Prony-based "big prime" technique, which acts over a single large finite field, or the more recently-proposed "small primes" technique, which reduces the unknown sparse polynomial to many low-degree dense polynomials. While the latter technique has not yet reached the same theoretical efficiency as Prony-based methods, it has an obvious potential for parallelization. We present a heuristic "small primes" interpolation algorithm and report on a low-level C implementation using FLINT and MPI.
Based on the computation of a superset of the implicit support, implicitization of a parametrically given hyper-surface is reduced to computing the nullspace of a numeric matrix. Our approach exploits the sparseness of the given parametric equations and of the implicit polynomial. In this work, we study how this interpolation matrix can be used to reduce some key geometric predicates on the hyper-surface to simple numerical operations on the matrix, namely membership and sidedness for given query points. We illustrate our results with examples based on our Maple implementation.
Given a straight-line program whose output is a polynomial function of the inputs, we present a new algorithm to compute a concise representation of that unknown function. Our algorithm can handle any case where the unknown function is a multivariate polynomial, with coefficients in an arbitrary finite field, and with a reasonable number of nonzero terms but possibly very large degree. It is competitive with previously known sparse interpolation algorithms that work over an arbitrary finite field, and provides an improvement when there are a large number of variables.
We present new techniques for reducing a multivariate sparse polynomial to a univariate polynomial. The reduction works similarly to the classical and widely-used Kronecker substitution, except that we choose the degrees randomly based on the number of nonzero terms in the multivariate polynomial, that is, its sparsity. The resulting univariate polynomial often has a significantly lower degree than the Kronecker substitution polynomial, at the expense of a small number of term collisions. As an application, we give a new algorithm for multivariate interpolation which uses these new techniques …
We give a new probabilistic algorithm for interpolating a "sparse" polynomial f given by a straight-line program. Our algorithm constructs an approximation f* of f, such that their difference probably has at most half the number of terms of f, then recurses on their difference. Our approach builds on previous work by Garg and Schost (2009), and Giesbrecht and Roche (2011), and is asymptotically more efficient in terms of the total cost of the probes required than previous methods, in many cases.
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. To interpolate a polynomial $f\in F_q[x_1,\dots,x_n]$ with a partial degree bound $D$ and a term bound $T$, our new algorithm costs $O^\thicksim(nT\log ^2q+nT\sqrt{D}\log q)$ bit operations and uses $2(n+1)T$ probes to the black box. If $q\geq O(nT^2D)$, it has constant s…
We present the main improvements and new features in version $\texttt{2.0}$ of the open-source $\texttt{C++}$ library $\texttt{FireFly}$ for the interpolation of rational functions. This includes algorithmic improvements, e.g. a hybrid algorithm for dense and sparse rational functions and an algorithm to identify and remove univariate factors. The new version is applied to a Feynman-integral reduction to showcase the runtime improvements achieved. Moreover, $\texttt{FireFly}$ now supports parallelization with $\texttt{MPI}$ and offers new tools like a parser for expressions or an executable fo…
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 $f(x)$…
We present sparse interpolation algorithms for recovering a polynomial with $\le B$ terms from $N$ evaluations at distinct values for the variable when $\le E$ of the evaluations can be erroneous. Our algorithms perform exact arithmetic in the field of scalars $\mathsf{K}$ and the terms can be standard powers of the variable or Chebyshev polynomials, in which case the characteristic of $\mathsf{K}$ is $\ne 2$. Our algorithms return a list of valid sparse interpolants for the $N$ support points and run in polynomial-time. For standard power basis our algorithms sample at $N = \lfloor \frac{4}{3…
Given a "black box" function to evaluate an unknown rational polynomial f in Q[x] at points modulo a prime p, we exhibit algorithms to compute the representation of the polynomial in the sparsest shifted power basis. That is, we determine the sparsity t, the shift s (a rational), the exponents 0 <= e1 < e2 < ... < et, and the coefficients c1,...,ct in Q\{0} such that f(x) = c1(x-s)^e1+c2(x-s)^e2+...+ct(x-s)^et. The computed sparsity t is absolutely minimal over any shifted power basis. The novelty of our algorithm is that the complexity is polynomial in the (sparse) representation size, and in…
We present algorithms performing sparse univariate polynomial interpolation with errors in the evaluations of the polynomial. Based on the initial work by Comer, Kaltofen and Pernet [Proc. ISSAC 2012], we define the sparse polynomial interpolation codes and state that their minimal distance is precisely the length divided by twice the sparsity. At ISSAC 2012, we have given a decoding algorithm for as much as half the minimal distance and a list decoding algorithm up to the minimal distance. Our new polynomial-time list decoding algorithm uses sub-sequences of the received evaluations indexed b…
