Finite relative determination and the Artin-Rees lemma (Q1047535)

The authors study finite determinacy properties of quasi-homogeneous function germs in two variables of weights \( ({1\over 3},{1\over 6}) \) and degree \(1\), with respect to the group of germs of diffeomorphisms of the form \((\alpha x+\beta y^2 +h_1(x,y), \delta y+h_2(x,y)) \) with \( h_1\in \underline{m}_2^3 \) and \( h_2\in \underline{m}_2^2 \) and \( \alpha\delta\neq 0 \). They compare two families of germs with different determinacy numbers. They also discuss a variant of the classic Artin-Rees lemma for radical ideals in the ring of smooth germs.