Jacobi's bound for systems of algebraic differential equations

This review paper is devoted to the Jacobi bound for systems of partial differential polynomials. We prove the conjecture for the system of $n$ partial differential equations in $n$ differential variables which are independent over a prime differential ideal $\mathfrak p$. On the one hand, this generalizes our result about the Jacobi bound for ordinary differential polynomials independent over a prime differential ideal $\mathfrak p$ and, on the other hand, the result by Tomasovic, who proved the Jacobi bound for linear partial differential polynomials.