On reduced Arakelov divisors of real quadratic fields

We generalize the concept of reduced Arakelov divisors and define \(C\)-reduced divisors for a given number \(C \geq 1\). These \(C\)-reduced divisors have remarkable properties which are similar to the properties of reduced ones. In this paper, we describe an algorithm to test whether an Arakelov divisor of a real quadratic field \(F\) is \(C\)-reduced in time polynomial in \(\log|\Delta_F|\) with \(\Delta_F\) the discriminant of \(F\). Moreover, we give an example of a cubic field for which our algorithm does not work.