Valuation of an ideal in a two-dimensional regular local ring

Question

Let $f,g$ be two coprime elements in the ring $K[[x,y]]$, with $K$ a field. What is the smallest integer $n$ such that the inclusion of ideals $$(x^n)\subset (f,g)$$ holds in $K[[x,y]]$? Can we compute easily that integer? It's well known that $n$ is smaller or equal to the $x$-adic valuation of the $y$-resultant of the Weierstrass polynomials associated to $f$ and $g$. But can we do better in general? If yes, what is the geometrical interpretation of this integer in terms of the singularities of $f$ and $g$?

Answer 1

A few examples, not a general answer nor explanation in geometry (check my algebra):

1. f=y^a-x^b, g=y^c-x^d => n=min { ab/(a,c), ad/(a,c), bd/(b,d), b²c/((b,d)(a,b²c/(b,d))) }; ord resultant = min{a,c}min{b,d} (here, (a,c)=gcd(a,c) etc)

   f=y³-x, g=y²-x => n=1, ord resultant=2

   f=y-x³, g=y-x² => n=2, ord resultant=2

2. f=y^a-x^b, g=y^cx^d => n depends on relative sizes of b,c and a,d

   f=y³-x, g=y³x => n=2, ord resultant=6