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On reduced Arakelov divisors of real quadratic fields

Ha Thanh Nguyen Tran — 2016

Acta Arithmetica

We generalize the concept of reduced Arakelov divisors and define C-reduced divisors for a given number C ≥ 1. These C-reduced divisors have remarkable properties, similar to the properties of reduced ones. We describe an algorithm to test whether an Arakelov divisor of a real quadratic field F is C-reduced in time polynomial in $l o g | Δ_{F} |$ with $Δ_{F}$ the discriminant of F. Moreover, we give an example of a cubic field for which our algorithm does not work.

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