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Zbl 0982.13011
Neelon, Tejinder S.
On solutions to formal equations. (English)
[J] Bull. Belg. Math. Soc. - Simon Stevin 7, No.3, 419-427 (2000). ISSN 1370-1444

Summary: Let $\overline k$ be a field of characteristic zero equipped with an absolute value $|\cdot|$. Let $\varphi_1({\bold x}, {\bold y})= \varphi_2 ({\bold x},{\bold y}) =\cdots= \varphi_l ({\bold x},{\bold y})=0$ be a system of formal power series equations in variables ${\bold x}= (x_1, \dots,x_n)$, ${\bold y}=(y_1, \dots,y_m)$ with coefficients in $\overline k$. The notion of $\{M_k\}$-summability of formal power series is defined relative to a sequence $\{M_k\}^\infty_{k=0}$ of positive real numbers. Under certain Jacobian conditions on the $\varphi_i$'s, it is shown the $\{M_k\}$-summability of the $\varphi_i$'s implies $\{M_k\}$-summability of any of its formal power series solutions ${\bold y}={\bold f}({\bold x})$. In particular, if the $\varphi_i$'s are convergent, then so are the formal solutions. This result generalizes the author's earlier work on formal solutions of systems of analytic equations [{\it T. S. Neelon}, Proc. Am. Math. Soc. 125, No. 9, 2531-2535 (1997; Zbl 0890.32004)].

MSC 2000:: *13F25 Formal power series rings
40G99 Special methods of summability

Keywords: summability of formal power series

Citations: Zbl 0890.32004

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