Resumen de Counting exceptional units

Gerhard Niklasch

• The number of solutions of the "unit equation" $x + y = 1$ in units of (the ring of integers of) an algebraic number field of degree $n$ and unit rank $r$ is known to be bounded above by an exponential function of $n$ and $r$, but the best known lower bounds are only polynomial in $n$, and the true counts have been computed only in a few cases.\newline We will present recently computed solution counts in number fields of unit rank $r\leq 5$, leading to a tentative formula for the largest number of solutions attained by at least one field of given signature. The formula agrees with the Stewart heuristic, predicting about exp$(r^{2/3+o(1)})$ solutions. These counts are dominated by "small" solutions, whereas the smaller number of solutions which can be attained infinitely often by fields of a fixed signature hinges on the "large" ones.