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.