Sparse Polynomial Interpolation and Division in Soft-linear Time

Abstract

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. ese 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 an integer polynomial ( ) given a way to evaluate ( ) mod for any chosen integers and . is algorithm has nearly-optimal bit complexity, meaning that the total bit-length of the probes, as well as the computational running time, is so ly linear (ignoring logarithmic factors) in the bit-length of the resulting sparse polynomial. To our knowledge, this is the first sparse interpolation algorithm with so -linear bit complexity in the total output size. For integer polynomials, the best previously known results have at least a cubic dependency on the bit-length of the exponents. * In this work, we do not consider the case of unbalanced bit lengths, where the differing sizes of each coefficient and exponent are considered in the complexity.