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1. CJM 2014 (vol 67 pp. 286)
A Skolem-Mahler-Lech Theorem for Iterated Automorphisms of $K$-algebras This paper proves a commutative algebraic extension
of a generalized Skolem-Mahler-Lech theorem due to the first
author.
Let $A$ be a finitely generated commutative $K$-algebra
over a field of characteristic $0$, and let $\sigma$ be
a $K$-algebra automorphism of $A$.
Given ideals $I$ and $J$ of $A$, we show that
the set $S$ of integers $m$ such that
$\sigma^m(I) \supseteq J$ is a finite union of
complete doubly infinite arithmetic progressions in $m$, up to
the addition of a finite set.
Alternatively, this result states that for an affine scheme
$X$ of finite type over $K$,
an automorphism $\sigma \in \operatorname{Aut}_K(X)$, and $Y$ and $Z$
any two closed subschemes of $X$, the set
of integers $m$ with $\sigma^m(Z ) \subseteq Y$ is as above.
The paper presents examples
showing that this result may fail to hold if the affine scheme
$X$ is
not of finite type, or if $X$ is of finite type but the field
$K$ has positive characteristic.
Keywords:automorphisms, endomorphisms, affine space, commutative algebras, Skolem-Mahler-Lech theorem Categories:11D45, 14R10, 11Y55, 11D88 |
2. CJM 2013 (vol 66 pp. 844)
Multidimensional Vinogradov-type Estimates in Function Fields Let $\mathbb{F}_q[t]$ denote the polynomial ring over the finite
field $\mathbb{F}_q$.
We employ Wooley's new efficient congruencing method to prove
certain multidimensional Vinogradov-type estimates in $\mathbb{F}_q[t]$.
These results allow us to apply a variant of the circle method
to obtain asymptotic formulas for a system connected to the problem
about linear spaces lying on hypersurfaces defined over $\mathbb{F}_q[t]$.
Keywords:Vinogradov's mean value theorem, function fields, circle method Categories:11D45, 11P55, 11T55 |
3. CJM 2001 (vol 53 pp. 897)
On Some Exponential Equations of S.~S.~Pillai In this paper, we establish a number of theorems on the classic
Diophantine equation of S.~S.~Pillai, $a^x-b^y=c$, where $a$, $b$ and
$c$ are given nonzero integers with $a,b \geq 2$. In particular, we
obtain the sharp result that there are at most two solutions in
positive integers $x$ and $y$ and deduce a variety of explicit
conditions under which there exists at most a single such solution.
These improve or generalize prior work of Le, Leveque, Pillai, Scott
and Terai. The main tools used include lower bounds for linear forms
in the logarithms of (two) algebraic numbers and various elementary
arguments.
Categories:11D61, 11D45, 11J86 |