The authors consider the problem of reconstructing (i.e., interpolating) a t-sparse multivariate polynomial given a black box which will produce the value of the polynomial for any value of the arguments. It is shown that, if the polynomial has coefficients in a finite field GF[q] and the black box can evaluate the polynomial in the field GF[qr2g,tnt+37], where n is the number of variables, then there is an algorithm to interpolate the polynomial in O(log (nt)) boolean parallel time and O(n2t log nt) processors. 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 with the fact that efficient interpolation using a black box that only evaluates the polynomial at points in GF[q] is

We consider the task of reconstructing algebraic functions given by black boxes. Unlike traditional settings, we are interested in black boxes which represent several algebraic functions-f1;:::;fk, where at each inputx, the box arbitrarily chooses a subset off1(x);:::;fk(x)to output. We show how to reconstruct the functionsf1;:::;fkfrom the black box. This allows us to group the sample points into sets, such that for each set, all outputs to points in the set are from the same algebraic function. Our methods are robust in the presence of errors in the black box. Our model and techniques can be applied in the areas of computer vision, machine learning, curve fitting and polynomial approximation, self-correcting programs and bivariate polynomial factorization. 1

. We consider the problem of interpolating sparse multivariate polynomials from their values. We discuss two algorithms for sparse interpolation, one due to Ben-Or and Tiwari (1988) and the other due to Zippel (1988). We present efficient algorithms for finding the rank of certain special Toeplitz systems arising in the Ben-Or and Tiwari algorithm and for solving transposed Vandermonde systems of equations, the use of which greatly improves the time complexities of the two interpolation algorithms. 1. Introduction We consider the problem of interpolating a multivariate polynomial over a field of characteristic zero from its values at several points. While techniques for interpolating dense polynomials have been known for a long time (e.g., Lagrangian interpolation formula for univariate polynomials), and probabilistic algorithms for interpolating sparse multivariate polynomials have existed since 1979 (Zippel 1979, 1988), until recently no algorithm was known to interpolate sparse mul...

Abstract not found

We present a randomized algorithm that interpolates a sparse polynomial in polynomial time in the bit complexity model. The algorithm can be also applied to approximate polynomials that can be approximated by sparse polynomials (the approximation is in the L_2 norm).

We consider the problem of sparse interpolation of an approximate multivariate black-box polynomial in floating-point arithmetic. That is, both the inputs and outputs of the black-box polynomial have some error, and all numbers are represented in standard, fixed-precision, floating point arithmetic. By interpolating the black box evaluated at random primitive roots of unity, we give efficient and numerically robust solutions. We note the similarity between the exact Ben-Or/Tiwari sparse interpolation algorithm and the classical Prony’s method for interpolating a sum of exponential functions, and exploit the generalized eigenvalue reformulation of Prony’s method. We analyze the numerical stability of our algorithms and the sensitivity of the solutions, as well as the expected conditioning achieved through randomization. Finally, we demonstrate the effectiveness of our techniques in practice through numerical experiments and applications. 1.

In this work we study two, seemingly unrelated, notions. Locally decodable codes (LDCs) are codes that allow the recovery of each message bit from a constant number of entries of the codeword. Polynomial identity testing (PIT) is one of the fundamental problems of algebraic complexity: we are given a circuit computing a multivariate polynomial and we have to determine whether the polynomial is identically zero. We improve known results on LDCs and on polynomial identity testing and show a relation between the two notions. In particular we obtain the following results: (1) We show that if E: F n ↦ → F m is a linear LDC with two queries, then m = exp(Ω(n)). Previously this was known only for fields of size ≪ 2 n [O. Goldreich et al., Comput. Complexity, 15 (2006), pp. 263–296]. (2) We show that from every depth 3 arithmetic circuit (ΣΠΣ circuit), C, with a bounded (constant) top fan-in that computes the zero polynomial, one can construct an LDC. More formally, assume that C is minimal (no subset of the multiplication gates sums to zero) and simple (no linear function appears in all the multiplication gates). Denote by d the degree of the polynomial computed by C and by r the rank of the linear functions appearing in C. Then we can construct a linear LDC with two queries that encodes messages of length r/polylog(d) by codewords of length O(d). (3) We prove a structural theorem for ΣΠΣ circuits, with a bounded top fan-in, that

: Many current geometric modeling systems use the rational parametric form to represent surfaces. Although the parametric representation is useful for tracing, rendering and surface fitting, many operations like surface intersection desire one of the surfaces to be represented implicitly. Moreover, the implicit representation can be used for testing whether a point lies on the surface boundary and to represent an object as a semi-algebraic set. Previously resultants and Grobner basis have been used to implicitize parametric surfaces. In particular, different formulations of resultants have been used to implicitize tensor product surfaces and triangular patches and in many cases the resulting expression contains an extraneous factor. The separation of these extraneous factors can be a time consuming task involving multivariate factorization. Furthermore, these algorithms fail altogether if the given parametrization has base points. In this paper we present an algorithm to implicitize pa...

A probabilistic strategy, early termination, enables di#erent interpolation algorithms to adapt to the degree or the number of terms in the target polynomial when neither is supplied in the input. In addition to dense algorithms, we implement this strategy in sparse interpolation algorithms. Based on early termination, racing algorithms execute simultaneously a dense and a sparse algorithm. The racing algorithms can be embedded as the univariate interpolation substep within Zippel's multivariate method. In addition, we experimentally verify some heuristics of early termination, which make use of thresholds and post-verification. Key words: Early termination, sparse polynomial, black box polynomial, interpolation, sparse interpolation, randomized algorithm, Chebyshev basis, Pochhammer basis, racing two algorithms, Zippel's algorithm, Ben-Or's and Tiwari's algorithm. Email addresses: kaltofen@math.ncsu.edu (Erich Kaltofen), ws2lee@scg.uwaterloo.ca (Wen-shin Lee).

We study the learnability of multiplicity automata in Angluin’s exact learning model, and we investigate its applications. Our starting point is a known theorem from automata theory relating the