On the scalar rational interpolation problem (Q1356628)

This paper addresses the scalar interpolation problem for an arbitrary sequence of points over a field. The main aim of the paper is to give an algorithm extending earlier ones by the author and others for the partial realization (solving the key equation of error correcting codes) and classical interpolation problems, to the general interpolation problem. Instead of solving the general interpolation problem directly, a module-theoretic formulation of the weak interpolation problem is found and reduced Gröbner bases of the free module of rank 2 over the polynomial ring in one variable are employed to parametrize weak interpolations. A recursive algorithm for the determination of reduced Gröbner bases is given. Using the classification of weak interpolations, the main theorem determines a parametrization of all minimal complexity rational functions \(\frac{a(x)}{b(x)}\) interpolating an arbitrary sequence of points, where complexity is measured in terms of \(\max\{\deg(a), \deg(b)+r\}\), and \(r\) is an arbitrary integer.