thank you. so you are saying that this definition of tangent cone is "wrong". is there a "right" definition such that the answer to my question becomes yes? is the "wrong" tangent cone of a complete intersection perhaps always a set-theoretic complete intersection?

I will look at your related question, but I can't promise anything.

@guest: one way to generalize realrootedness to more variables is the concept of hyperbolic polynomials. you can define the interlacing property for them too (for example arxiv.org/abs/1212.6696). and most of what I've written should extend to that case.

@user6818: I think the proof should work with similar techniques as in this paper. at the moment I have no reference.

thank you. is it true that since $X$ is Cohen-Macaulay, I just have to check whether $S$ localized at the irrelevant ideal is Cohen-Macaulay? And is it correct that $X$ is arithmetically Cohen-Macaulay if and only if the affine cone $\hat{X}$ is Cohen-Macaulay?