Let \(C[[X_ 1,...,X_ n]]\) be the ring of formal power series and \({\mathfrak M}\) be the maximal ideal of this ring. Recall that \(f\in {\mathfrak M}\) is quasihomogeneous of type \((d;r)\), \(r=(r_ 1,...,r_ n)\in Q^ n\), if each monomial \(X_ 1^{i_ 1}...X_ n^{i_ n}\) of f has the generalized degree d or \(\infty\), i.e. \(r_ 1i_ 1+...+r_ ni_ n=d\) respectively \(r_ 1i_ 1+...+r_ ni_ n=0\). Denote by \(\Delta f\) the Jacobi ideal, namely the ideal spanned by the partial derivatives of \(f\). A formal power series \(f\in {\mathfrak M}\) is non-degenerate if there exists a natural number \(p\) such that \({\mathfrak M}\supset \Delta f\supset {\mathfrak M}^ p\). If \(f\) is a nondegenerate quasihomogeneous series of type \((d;r)\), the number of basis monomials of generalized degree \(\geq d\) is called the inner modality of \(f\) and one denotes by \(m(f)\).
V. I. Arnol’d [Russ. Math. Surv. 29, 10–50 (1974); translation from Usp. Mat. Nauk 29, No.2(176), 11-49 (1974; Zbl 0298.57022), Invent. Math. 35, 87–109 (1976; Zbl 0336.57022)] has begun the classification of non-degenerate quasihomogeneous functions with \(m(f)=0,1\). This classification was continued by E. Yoshinaga and M. Suzuki [Invent. Math. 55, 185–206 (1979; Zbl 0406.58008)] and by the authors of this paper [Preprint Nr. 53, Humboldt-Univ. Berlin (1983)], [Rev. Cienc. Mat. 4, No. 3, 205–216 (1983; Zbl 0559.05004)] in the cases \(m(f)=2,3,4,5.\)
In the present paper the authors find all normal forms on nondegenerate quasihomogeneous polynomials f with \(m(f)=6\). There are also given all normal forms of nondegenerate quasihomogeneous polynomials which are boundaries of nondegenerate quasihomogeneous polynomials with \(m=6\). In the last part of the paper the adjacencies among these new classes are studied.