On an arithmetical property of the normalization of regular systems of weights (Q1201456)

The weights of a weighted homogeneous polynomial \(f\in\mathbb{C}[x,y,z]\) with isolated singularity form a regular system of weights: four positive integer \((a,b,c;h)\), whose characteristic function \[ \chi(T)=T^{- h}{(T^ h-T^ a)(T^ h-T^ b)(T^ h-T^ b)\over(T^ a-1)(T^ b- 1)(T^ c-1)} \] is regular on \(\mathbb{C}\backslash 0\). Write \(\chi(T)=\sum a_ nT^ n\). The coefficient \(a_ 0\) is equal to the genus of the central curve in the resolution of \(f\) (there are also \(r\) branches), and \(m_ 0:=a_{h-a-b-c}\) is the dimension of the degree \(h\) part of the Jacobi ring of \(f\). This paper gives an arithmetical proof (i.e. without singularity theory) of the inequality \(m_ 0\geq r-3+a_ 0\), with equality if \(a_ 0=0\).