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# Smooth Polynomial Solutions to a Ternary Additive Equation

Published:2017-10-05
Printed: Feb 2018
• Junsoo Ha,
Korea Institute for Advanced Study, Seoul, Republic of Korea
 Format: LaTeX MathJax PDF

## Abstract

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
 MSC Classifications: 11T55 - Arithmetic theory of polynomial rings over finite fields 11D04 - Linear equations 11L07 - Estimates on exponential sums 11T23 - Exponential sums

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