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Search: MSC category 11T55 ( Arithmetic theory of polynomial rings over finite fields )

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1. CJM Online first

Ha, Junsoo
Smooth Polynomial Solutions to a Ternary Additive Equation
Let $\mathbf{F}_{q}[T]$ be the ring of polynomials over the finite field of $q$ elements, and $Y$ be a large integer. We say a polynomial in $\mathbf{F}_{q}[T]$ is $Y$-smooth if all of its irreducible factors are of degree at most $Y$. We show that a ternary additive equation $a+b=c$ over $Y$-smooth polynomials has many solutions. As an application, if $S$ is the set of first $s$ primes in $\mathbf{F}_{q}[T]$ and $s$ is large, we prove that the $S$-unit equation $u+v=1$ has at least $\exp(s^{1/6-\epsilon}\log q)$ solutions.

Keywords:smooth number, polynomial over a finite field, circle method
Categories:11T55, 11D04, 11L07, 11T23

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

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