The Jacobi Bound Conjecture for differential algebraic varieties

Let $K$ be a differential field, and let $\Sigma$ be a differential algebraic variety defined by $u_1,u_2,\ldots,u_n\in K\lbrace x_1,\ldots,x_n\rbrace$. Let $\Sigma_1$ be an irreducible component of $\Sigma$ with finite Krull dimension. For the tuple $(u_1,\ldots,u_n)$, define its Jacobi number by

$$J(u_1,\ldots,u_n)=\operatorname{max}_{\sigma\in S_n}\sum_{i=1}^n\operatorname{ord}_{x_i}^{\partial}(u_{\sigma(i)}).$$

Jacobi Bound Conjecture. The absolute dimension of $\Sigma_1$ satisfies

$$\operatorname{dim}(\Sigma_1)\leq J(u_1,u_2,\ldots,u_n).$$

This conjecture proposes a bound on the dimension of an irreducible finite-dimensional component of a differential algebraic variety in terms of the Jacobi number of its defining differential equations. It is attributed in the source to Ritt and Kolchin; its resolution status is not established by the supplied material.