Rational Univariate Representation （RUR） of zero-dimensional ideals is used to describe the zeros of zero-dimensional ideals and RUR has been studied extensively. In 1999, Roullier proposed an efficient algorithm to compute RUR of zero-dimensional ideals. In this paper, we will present a new algorithm to compute Polynomial Univariate Representation （PUR） of zero-dimensional ideals. The new algorithm is based on some interesting properties of Grobner basis. The new algorithm also provides a method for testing separating elements.
Insa and Pauer presented a basic theory of Grobner basis for differential operators with coefficients in a commutative ring in 1998, and a criterion was proposed to determine if a set of differential operators is a GrSbner basis. In this paper, we will give a new criterion such that Insa and Pauer＇s criterion could be concluded as a special case and one could compute the Grobner basis more efficiently by this new criterion.
