Title: Simultaneous Rational Function Reconstruction and applications to Algebraic Coding Theory.
Abstract: The simultaneous rational function reconstruction (SRFR) is the
problem of reconstructing a vector of rational functions with the same denominator
given its evaluations (or more generally given its remainders modulo different
polynomials).
The peculiarity of this problem consists in the fact that the common denominator
constraint reduces the number of evaluation points needed to guarantee the existence
of a solution, possibly losing the uniqueness. One of the main contributions presented
in this talk consists in the proof that uniqueness is guaranteed for almost all instances
of this problem.
This result was obtained by elaborating some other contributions and techniques
derived by the applications of SRFR, from the polynomial linear system solving to the
decoding of Interleaved Reed-Solomon codes. In this talk it is also presented another
application of the SRFR problem, concerning the problem of constructing fault-tolerant
algorithms: algorithms resilients to computational errors.
These algorithms are constructed by introducing redundancy and using error correcting
codes tools to detect and possibly correct errors which occur during computations.
In this application context, we improve an existing fault-tolerant technique for
polynomial linear system solving by interpolation-evaluation, by focusing on the related
SRFR.
Contains joint works with Eleonora Guerrini and Romain Lebreton.
