Abstract
This dissertation deals with a Computer Algebra problem which has significant consequencesin Algebraic Coding Theory and Error Correcting Codes: the simultaneous rationalfunction reconstruction. Indeed, an accurate analysis of this problem leads to interestingresults in both these scientific domains.More precisely, the simultaneous rational function reconstruction is the problem of reconstructinga vector of rational functions with the same denominator given its evaluations(or more generally given its remainders modulo different polynomials). The peculiarity ofthis problem consists in the fact that the common denominator constraint reduces the numberof evaluation points needed to guarantee the existence of a solution, possibly losing theuniqueness. One of the main contribution of this work consists in the proof that uniquenessis guaranteed for almost all instances of this problem.This result was obtained by elaborating some other contributions and techniques derivedby the applications of SRFR, from the polynomial linear system solving to the decoding ofInterleaved Reed-Solomon codes.In this work, we will also study and present another application of the SRFR problem,concerning the problem of constructing fault-tolerant algorithms: algorithms resilientsto computational errors. These algorithms are constructed by introducing redundancy andusing error correcting codes tools to detect and possibly correct errors which occur duringcomputations. In this application context, we improve an existing fault-tolerant techniquefor polynomial linear system solving by interpolation-evaluation, by focusing on the SRFRproblem related to it.