# Differential criterion for regular sequences

Let $$R$$ be a subalgebra of the polynomial ring $$\mathbb{C}[X_1,\ldots,X_n]$$. Suppose $$\theta=(\theta_1,\ldots,\theta_p)$$ is a sequence of of elements in $$R$$. If I want to test for their algebraic independence, I have a simple differential criterion: compute the Jacobian matrix $$(\frac{\partial\theta_i}{\partial x_j})$$ and find at least one maximal minor which does not vanish identically as a polynomial in the $$x$$'s. Now I am interested in the stronger property of $$\theta$$ being an $$R$$-regular sequence. Is there a similar criterion I could use? I don't mind if it is complicated involving higher derivatives, jets and whatnot, but it would have to be rather concrete and explicit. The motivation for my question comes from my recent work on rings of invariants.

• Whether $\theta_i$s are algebraically independent did not depend on $R$, only in the polynomial ring. But, regular sequence is very much dependent on $R$, so conditions as elements of the polynomial rings will be insufficient. May be as an example, consider $\mathbb{C}[x^4, x^3y,xy^3,y^4]=R$. Then, $x^4, y^4\in R$ do not form a regular sequence in $R$, but in the polynomial ring, they do. – Mohan Jun 5 at 21:42
• @Mohan: +1. Yes I know my question is a bit ill-formulated at the moment, precisely for the reason you mentioned. I guess one needs some info about the ring $R$ is but I would have to think more what that missing hypothesis for $R$ should be. If this helps, the example of $R$ I am interested in, is the ring of invariants of some reductive group like $SL_2$. – Abdelmalek Abdesselam Jun 5 at 22:08
• I should add: this is about how should $R$ be given? Giving $R$ in terms of generators is useless for my purposes. Maybe the image of a Reynolds operator, which in theory I know how to compute, is much better. – Abdelmalek Abdesselam Jun 5 at 22:11