On the scalar rational interpolation problem

A new technique is derived for determining a parametrization of all minimal complexity rational functions a(x)/b(x) interpolating an arbitrary sequence of points. Complexity is measured in terms of max{deg(a),deg(b) + r} where r is an arbitrary integer (so that r = 0 corresponds to the McMillan degree). Our construction uses Grobner bases of submodules of the free module of rank 2 over the polynomial ring in one variable and extends previous work on the key equation of error control coding theory.