I have the following problem (it arises in a calculation in Topological String Theory in the paper here that the author glosses over when evaluating a spectral sequence, but condenses down to this assertion):
Let $W \in \mathbb{C}[x_1, \dots, x_n]$ be a complex polynomial with nondegenerate critical points, i.e. at every point $x\in \mathbb{C}^n$ where all the (formal) partial derivatives $\partial_i W$ are $0$, the Hesse matrix of $W$ is nondegenerate (i.e. invertible). What the Author seems to use in a calculation is that then, the sequence $\partial_1 W, \dots, \partial_n W$ in $\mathbb{C}[x_1, \dots, x_n]$ is regular (which helps evaluate the Koszul cohomology of a certain chain complex). To spell this out, e.g. $\partial_1 W$ should not be a zero divisor in $\mathbb{C}[x_1, \dots, x_n]/(\partial_2 W, \dots, \partial_n W)$, and the same for all other derivatives.
In all examples I tried out, this seemed to be true; however I can't find an actual proof of this (maybe it is a standard fact of singularity theory?). Maybe it isn't even true, but I could show that the statement made in the above paper wouldn't hold then, so it should better be...