Counting exceptional units
نویسنده
چکیده
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. We will present recently computed solution counts in number fields of unit rank r ≤ 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 ) 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.
منابع مشابه
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تاریخ انتشار 2002