Decomposing polynomial sets into simple sets over finite fields: The positive-dimensional case

This paper presents an algorithm for decomposing any positive-dimensional polynomial set into simple sets over an arbitrary finite field. The algorithm is based on some relationship established between simple sets and radical ideals, reducing the decomposition problem to the problem of computing the radicals of certain ideals. In addition to direct application of the algorithms of Matsumoto and Kemper, the algorithm of Fortuna and others is optimized and improved for the computation of radicals of special ideals. Preliminary experiments with an implementation of the algorithm in Maple and Singular are carried out to show the effectiveness and efficiency of the algorithm.