Abstract: The interaction between greatest common divisors and Diophantine equations provides a way to study sums of powers. By considering classical problems in number theory, we trace the passage to gcd estimates and zero-counting results for $S$-units over function fields. This allows us to study Catalan-type equations $f_1x^a+f_2y^b+f_3z^c=1$ over the function field of a curve. When $a$ and $b$ are sufficiently large, we describe a height bound for solutions outside explicit exceptional families. We will also discuss a higher-dimensional analogue in which nonconstant solutions are constrained to a proper algebraic subset. The emphasis will be on motivation and the main statements rather than proof details.
Il seminario sarà presentato nell'aula C308. Per ulteriori informazioni, si può contattare amos.turchet@uniroma3.it.
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