Dear Ansgar,
>  gap> EuclideanDegree(0*x^0);
>  -1
>  Are these answers correct ?

In an euclidean ring R the euclidean degree is defined *only* for the
non-zero elements of R. So one should *not* apply 'EuclideanDegree'
to the zero element of an euclidean ring, the behaviour is undefined
in this case.

gap> Degree(0*x^0);
-1

As there is no "-infty", -1 is used. This is convenient in polynomial
rings and hopefully not too confusing in laurent polynomial rings.

best wishes
Frank