Jacobi’s bound for the order of systems of first order differential equations

Abstract:

Let ${A_1}, \ldots ,{A_n}$ be a system of differential polynomials in the differential indeterminates ${y^{(1)}}, \ldots ,{y^{(n)}}$, and let $\mathcal {M}$ be an irreducible component of the differential variety $\mathcal {M}({A_1}, \ldots ,{A_n})$. If $\dim \mathcal {M} = 0$, there arises the question of securing an upper bound for the order of $\mathcal {M}$ in terms of the orders ${r_{ij}}$ of the polynomials ${A_i}$ in ${y^{(j)}}$. It has been conjectured that the Jacobi number \[ J = J({r_{ij}}) = \max \{\sum \limits _{i = 1}^n {{r_{i{j_i}}}} :{j_1}, \ldots ,{j_n}{\text { is a permutation of 1,}} \ldots ,n \}\] provides such a bound. In this paper $J$ is obtained as a bound for systems consisting of first order polynomials. Differential kernels are employed in securing the bound, with the theory of kernels obtained in a manner analogous to that of difference kernels as given by R. M. Cohn.