Georgia Institute of Technology College of Sciences

We present algorithms for performing sparse univariate polynomial interpolation with errors in the evaluations of the polynomial. Our interpolation algorithms use as a substep an algorithm that originally is by R. Prony from the French Revolution (Year III, 1795) for interpolating exponential sums and which is rediscovered to decode digital error correcting BCH codes over finite fields (1960). Since Prony's algorithm is quite simple, we will give a complete description, as an alternative for Lagrange/Newton interpolation for sparse polynomials. When very few errors in the evaluations are permitted, multiple sparse interpolants are possible over finite fields or the complex numbers, but not over the real numbers. The problem is then a simple example of list-decoding in the sense of Guruswami-Sudan. Finally, we present a connection to the Erdoes-Turan Conjecture (Szemeredi's Theorem). This is joint work with Clement Pernet, Univ. Grenoble.