Abstract view
Smooth Polynomial Solutions to a Ternary Additive Equation


Published:20171005
Printed: Feb 2018
Junsoo Ha,
Korea Institute for Advanced Study, Seoul, Republic of Korea
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.