This paper deals with the polynomial linear system solving with errors (PLSwE) problem. More specifically, we solve linear systems with univariate polynomial coefficients via an evaluation-interpolation technique assuming that errors can occur before the interpolation step. In this framework, the number of evaluations needed to recover the solution depends on the parameters of the linear system (degrees, size) and on the number of errors.

This paper focuses on the problem of reconstructing a vector of rational functions given some evaluations, or more generally given their remainders modulo different polynomials. The special case of rational functions sharing the same denominator, a.k.a. Simultaneous Rational Function Reconstruction (SRFR), has many applications from linear system solving to coding theory, provided that SRFR has a unique solution. The number of unknowns in SRFR is smaller than for a general vector of rational function. This allows one to reduce the number of evaluation points needed to guarantee the existence of a solution, possibly losing its uniqueness. In this work, we prove that uniqueness is guaranteed for a generic instance.

In this paper we present a new algorithm for Polynomial Linear System Solving (via evaluation/interpolation) with errors. In this scenario, errors can occur in the black box evaluation step. We improve the bound on the number of errors that we can correct, using techniques inspired by the decoding procedure of Interleaved Reed-Solomon Codes.