The Greenspan Bound for the Order of Differential Systems

Abstract

Let S be a system of ordinary differential polynomials in indeterminates ${y_1}, \ldots ,{y_n}$ and of order at most ${r_i}$ in ${y_i},1 \leqslant i \leqslant n$. It was shown by J. F. Ritt that if $\mathfrak {M}$ is a component of S of differential dimension 0, then the order of $\mathfrak {M}$ is at most ${r_1} + \ldots + {r_n}$. B. Greenspan improved this bound in the case that every component of S has differential dimension 0. (His work was carried out for difference equations, but is easily transferred to the differential case.) It is shown that the Greenspan bound is valid without this restriction.