Reduction of everywhere convergent power series with respect to Gröbner bases (Q1916423)

The formula for the division by a finite set \(F\) of polynomials of several variables \[ g=\sum_{f\in F}h_f\cdot f+g_{\text{red}} \] is generalized to the case of the ring \(E\) of everywhere convergent power series (\(g\), \(h_f\), \(g_{\text{red}}\) are power series then and, for a suitable norm \(|\cdot |_r\), \(|g_{\text{red}} |_r\leq|g|_r\)). The authors point out that the similar topic was investigated earlier by \textit{P. D. Djakov} and \textit{B. S. Mitiagin} [Stud. Math. 68, 85-104 (1980; Zbl 0434.46034)]. In the present article, however, Gröbner bases are used and, in the case, where \(F\) is the Gröbner basis of a polynomial ideal, uniqueness of \(g_{\text{red}}\), in a certain sense, is established. The used technique is applied to the proof of closedness of ideals generated by polynomials in \(E\) and to a simple proof for the affine version of \textit{J. P. Serre}'s graph theorem [GAGA, Ann. Inst. Fourier 6(1955/1956), 1-42 (1956; Zbl 0075.30401)].