On the convergence restrictions of divergent power series

Yu-Ching Hung, Purdue University

Abstract

It was proved that if ${F(X\sb1,\dots,}$ $X\sb{n})$ is a divergent power series in $n$ variables over $K$, a real valued field, then the convergence set $I\sbsp{n}{1}$ $-$ $Conv(F)$ is a 'small' $F\sb{\sigma}$ set (due to Abhyankar-Moh (1)). Conversely, given any 'small' $F\sb{\sigma}$ set $S$, there exists a divergent power series ${F(X\sb1,\dots,}$ $X\sb{n})$ with $I\sbsp{n}{1}$ $-$ $Conv(F)$ = $S$ (due to Sathaye (10)). In the classical case of real or complex numbers, $I\sbsp{n}{1}$ $-$ $Conv(F)$ must be of (logarithmic) capacity zero hence of measure zero if ${F(X\sb1,\dots,}$ $X\sb{n})$ is not convergent. In this work, we prove that if ${F(X\sb1,\dots,}$ $X\sb{n})$ is a divergent power series in ${X\sb1,\dots,}$ $X\sb{n}$ over $\IR$ or $\doubc$ and $m$ is any positive integer, then the convergence sets $I\sbsp{n}{m}$ $-$ $Conv(F)$ and $\Pi\sbsp{n}{m}$ $-$ $Conv(F)$ must be measure zero $F\sb{\sigma}$ sets.

Degree

Ph.D.

Advisors

Moh, Purdue University.

Subject Area

Mathematics

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