A zero-test and an interpolation algorithm for the shifted sparse polynomials

Abstract

Recall that a polynomial f ∈ F [X₁,...,X _n] is t-sparse, if f = ∑ α _IX^I contains at most t terms. In [BT 88], [GKS 90] (see also [GK 87] and [Ka 89]) the problem of interpolation of t-sparse polynomial given by a black-box for its evaluation has been solved. In this paper we shall assume that F is a field of characteristic zero. One can consider a t-sparse polynomial as a polynomial represented by a straight-line program or an arithmetic circuit of the depth 2 where on the first level there are multiplications with unbounded fan-in and on the second level there is an addition with fan-in t.

In the present paper we consider a generalization of the notion of sparsity, namely we say that a polynomial g(X₁,...,X _n) ∈ F[X₁,..., X _n] is shifted t-sparse if for a suitable nonsingular n x n matrix A and a vector B the polynomial g(A(X₁,...,X _n)^{T + B}) is t-sparse. One could consider g as being represented by a straight-line program of the depth 3 where on the first level (with the fan-in n + 1) a linear transformation A(X ₁,...,X _n) + B is computed. One could also consider a shifted t-sparse polynomial as t-sparse with respect to other coordinates (Y ₁,...,Y _n ^T=A(X)₁,...,X _n)^T + B.

We assume that a shifted t-sparse polynomial g is given by a black-box and the problem we consider is to construct a transformation A(X) ₁,...,X _n) + B. As the complexity of the designed below algorithm (see the Theorem in which we describe the variety of all possible A, B and the corresponding t-sparse representations of g(A(X ₁,...,X _n)^T + B)) depends on d ⁿ⁴ where d is the degree of g, we could first interpolate g within time d ^O(n) and suppose that g is given explicitly. It would be interesting to get rid of d in the complexity bounds as it is usually done in the interpolation of sparse polynomials ([BT 88], [GKS 90], [Ka 89]). The main technical tool we rely on is the criterium of t-sparsity based on Wronskian ([GKS 91], [GKS 92]), the latter criterium has a parametrical nature (so we can select t-sparse polynomials from a given parametrical family of polynomials) unlike the approach in [BT 88] using BCH-codes. We could directly consider (see the Theorem) the multivariate polynomials (section 3), but to make the exposition clearer before that we first study (see the proposition) the one-variable case (section 2). First at all we recall (section 1) the criterium of t-sparsity and based on it interpolation method for t-sparse multivariable polynomials.

In the last section 4 we design a zero-test algorithm for shifted t-sparse polynomials with the complexity independent on d.

Work partially done while visiting the Dept. of Computer Science, University of Bonn. On leave from the Steklov Mathematical Institute, Fontanka 27, St. Petersburg, 191011 Russia

Supported in part by Leibniz Center for Research in Computer Science, by the DFG Grant KA 673/4-1 and by the SERC Grant GR-E 68297