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Search: MSC category 11D45 ( Counting solutions of Diophantine equations )

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1. CJM Online first

Bell, Jason P.; Lagarias, Jeffrey C.
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)

Kuo, Wentang; Liu, Yu-Ru; Zhao, Xiaomei
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)

Bennett, Michael A.
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

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