Mathematical Research Letters
Volume 13 (2006)
One-parameter families of unit equations
Pages: 935 – 945
We study one-parameter families of $S$-unit equations of the form $f(t)u+g(t)v=h(t)$, where $f$, $g$, and $h$ are univariate polynomials over a number field, $t$ is an $S$-integer, and $u$ and $v$ are $S$-units. For many possible choices of $f$, $g$, and $h$, we are able to determine all but finitely many solutions to the corresponding one-parameter family of $S$-unit equations. The results are obtained as consequences of some recent results on integral points on surfaces.