A bound for the Rosenfeld–Gröbner algorithm

Abstract

We consider the Rosenfeld–Gröbner algorithm for computing a regular decomposition of a radical differential ideal generated by a set of ordinary differential polynomials in n indeterminates. For a set of ordinary differential polynomials F , let M ( F ) be the sum of maximal orders of differential indeterminates occurring in F . We propose a modification of the Rosenfeld–Gröbner algorithm, in which for every intermediate polynomial system F , the bound M ( F ) ⩽ ( n − 1 ) ! M ( F 0 ) holds, where F 0 is the initial set of generators of the radical ideal. In particular, the resulting regular systems satisfy the bound. Since regular ideals can be decomposed into characterizable components algebraically, the bound also holds for the orders of derivatives occurring in a characteristic decomposition of a radical differential ideal.