Ah, good catch. The way the question was phrased, it made it seem like even basic references would be helpful.

No, I'm more interested in a formula than an algorithm. If the answer turns out to be "No, except in simple cases there's no nice formula, but there's a Grobner basis algorithm that might work," the lack of a nice clean formula would be of interest to me as well. But, and forgive me if I'm being obtuse, it seems as if your answers are providing solutions on a case-by-case basis. If I was given two more random-looking power series (e.g., not monomials, not starting with $x+...$ or $y+...$), it's still unclear to me how to proceed.

+1, and this coming from someone who's definitely still in stage 1 in regards to not using the principle of the excluded middle. At least I now know what stage I'm in.

Very informative! +1

I should add on that the <i>point</i> of these things seems to be understanding special values of zeta functions at central values. I saw a talk of Ramachandran at one point where he gave an argument as to why these things should be related.

This is just vague recollections I have from having conversations about them with him, but I think that the idea is that $\mathbb{Q}(\frac{1}{2})$, for example, should be a motive whose tensor square is $\mathbb{Q}(1)$. So the whole business is trying to come up with what such things should look like.

I think I've included a better example of my confusion in the original post. Also, thanks for the reference above.

I think I'm still confused by this deformation idea. For a trivial non-monomial example, take $p=2$, $f=x+y$, and $g=y^2$. Then $I^2\subset (f,g)$ since $x^2=f^2-g$, $xy=xf-x^2$, and $yx=fx=x^2$. So we get $n=2$. But if we take an ordering where $y<x$, then the leading terms are $y$ and $y^2$, which gives $n=1$. Perhaps I'm misunderstanding what you're saying, but it seems unlikely to me that you'll be able to give an answer that depends only on the leading terms, no?