On the uniqueness of simultaneous rational function reconstruction

Abstract

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.