Jacobi’s bound for independent systems of algebraic partial differential equations

Abstract

In this paper we prove Jacobi’s bound for systems of n partial differential polynomials in n differential variables, which are independent over a prime differential ideal . This generalizes on the one hand our result (Kondratieva et al. in On Jacobi’s bound for systems of differential polynomials, algebra. Moscow University Press, Moscow, pp 79–85, 1982) about Jacobi’s bound for ordinary differential polynomials independent over a prime differential ideal and, on the other hand, the result by Tomasovic (A generalized Jacobi conjecture for arbitrary systems of algebraic differential equations. PhD thesis, Columbia University, 1976), who proposed a version of Jacobi’s bound for partial differential polynomials and proved it in the linear case. In Kondratieva et al. (Differential and difference dimension polynomials, Kluwer, Dordrecht, 1999) we gave another proof of this result by Tomasovic. The exposition in the present paper follows our proof in Kondratieva et al. (Differential and difference dimension polynomials, Kluwer, Dordrecht, pp 273–280, 1999).