On the existence of exponents prime to the Coxeter number (Q1102362)

This paper is a continuation of the study of the regular system of weights which was started in the author's previous paper [Adv. Stud. Pure Math. 8, 479-526 (1987; Zbl 0626.14028)]. A quadruple (a,b,c;h) of positive rational integers with \(\max (a,b,c)<h\) is called a system of weights, where a, b, c are called weights and h is called the Coxeter number. The system is called regular if \[ \chi (T):=T^{- h}\frac{(T\quad h-T\quad a)(T\quad h-T\quad b)(T\quad h-T\quad c)}{(T\quad a-1)(T\quad b-1)(T\quad c-1)} \] does not have a pole except \(T=0\). Then the coefficients of the Laurent series of \(\chi\) (T) at \(T=0\) are positive integers. So that there exist a finite number of integers \(m_ 1,...,m_{\mu}\), where \(\mu =(h-a)(h-b)(h-c)/abc\) such that \(\chi (T)=T^{m_ 1}+...+T^{m_{\mu}}.\) The integers \(m_ 1,...,m_{\mu}\) are called the exponents for (a,b,c;h). The system of weights (a,b,c;h) is called reduced if \(\gcd (a,b,c,h)=1.\) The main result in this paper is that for a reduced regular system of weights (a,b,c;h), 1, or -1 is an exponent for it and furthermore the multiplicity of 1 (resp. -1) as the exponents for (a,b,c;h) is equal to the number of positive integral solutions (u,v,w)\(\in {\mathbb{N}}^ 3 \)of the equation \(h+1=ua+vb+wc\) (resp. \(h-1=ua+vb+wc).\) Further, some applications for the study of two dimensional hypersurface singularities with C *-action are given.