Smooth Polynomial Solutions to aTernary Additive Equation

Ha, Junsoo

Canadian journal of mathematics, Tome 70 (2018) no. 1, pp. 117-141

Résumé

Let ${{\mathbf{F}}_{q}}[T]$ be the ring of polynomials over the finite field of $q$ elements and $Y$ 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 $\text{exp}\left( {{s}^{1/6-\in }}\,\text{log}\,\text{q} \right)$ solutions.

Détail

DOI : 10.4153/CJM-2017-023-x

Mots-clés : 11T55, 11D04, 11L07, 11T23, smooth number, polynomial over a finite field, circle method