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

Abstract

Let A 1 , … , A n {A_1}, \ldots ,{A_n} be a system of differential polynomials in the differential indeterminates y ( 1 ) , … , y ( n ) {y^{(1)}}, \ldots ,{y^{(n)}} , and let M \mathcal {M} be an irreducible component of the differential variety M ( A 1 , … , A n ) \mathcal {M}({A_1}, \ldots ,{A_n}) . If dim ⁡ M = 0 \dim \mathcal {M} = 0 , there arises the question of securing an upper bound for the order of M \mathcal {M} in terms of the orders r i j {r_{ij}} of the polynomials A i {A_i} in y ( j ) {y^{(j)}} . It has been conjectured that the Jacobi number \[ J = J ( r i j ) = max { ∑ i = 1 n r i j i : j 1 , … , j n is a permutation of 1, … , n } J = J({r_{ij}}) = \max \{\sum \limits _{i = 1}^n {{r_{i{j_i}}}} :{j_1}, \ldots ,{j_n}{\text { is a permutation of 1,}} \ldots ,n \} \] provides such a bound. In this paper J 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.