One-parameter families of unit equations

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.

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Published 1 January 2006