Apéry basis and polar invariants of plane curve singularities (Q804661)

Let C be an irreducible plane algebroid curve singularity defined by \(f\in K[[x,y]]\), K algebraically closed. Let \(\Gamma\) be the value semigroup of C, n be an integer and \(A_ n=\{0=a_ 0<a_ 1<...<a_{n- 1}\}=\{\min \Gamma \cap \{k+n{\mathbb{Z}}_{\geq 0}\}| k=0,...,n-1\}\) the Apéry basis of \(\Gamma\) relative to n. Let \(h\in K[[x,y]]\) such that the intersection multiplicity \((f\cdot h)=\dim_ KK[[x,y]]/(f,h)\) is an element of \(A_ n\), \(n=ord_ xf(x,0)\). Then the author proves a factorization theorem for h with certain precise combinatorial statements about numerical invariants of the factors and numerical invariants of f. This theorem applies to h being a generic polar of f and some results of \textit{Merle} and \textit{Ephraim} turn out to be special cases of the factorization theorem of the author.