Note on Diophantine inequality and Linear Artin Approximation over a local ring

Abstract

G. Rond [G. Rond, Approximation diophantienne dans les corps de séries en plusieurs variables, Ann. Institut Fourier 56 (2) (2006) 299–308. [10]] has proved Linear version of Artin Approximation theorem (LAA) and Diophantine inequality for a single homogeneous polynomial equation in two unknowns with coefficients in a formal (or convergent) power series ring over a field. M. Hickel and H. Ito, S. Izumi have generalized Rond's result to certain good local domains, independently, in 2008. This is a complementary Note to theirs. The most important point is that we can delete the equicharacteristic assumption in both papers. To cite this article: M. Hickel et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).