Sparse Rational Function Interpolation with Finitely Many Values for the Coefficients

TL;DR

This paper introduces new sparse interpolation algorithms for rational functions with integer coefficients, utilizing a strategic evaluation at a chosen integer to efficiently recover the functions, with near-optimal univariate and low-complexity multivariate methods.

Contribution

The paper presents novel sparse interpolation algorithms for rational functions that improve efficiency and complexity bounds, especially for multivariate cases.

Findings

Univariate interpolation algorithm is nearly optimal.

Multivariate interpolation has low complexity in T.

Data size grows exponentially with the number of variables.

Abstract

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.