  Loop invariants play a very important role in proving correctness of
programs. In this paper, we address the problem of generating invariants of
polynomial loop programs. We present a new approach, for generating polynomial
equation invariants of polynomial loop programs through computing vanishing
ideals of sample points. We apply rational function interpolation, based on
early termination technique, to generate invariants of loop programs with
symbolic initial values. Our approach avoids first-order quantifier elimination
and cylindrical algebraic decomposition(CAD). An algorithm for generating
polynomial invariants is proposed and some examples are given to illustrate the
algorithm. Furthermore, we demonstrate on a set of loop programs with symbolic
initial values that our algorithm can yield polynomial invariants with degrees
high up to 15.
