72 research outputs found
Computing heights on weighted projective spaces
In this note we extend the concept height on projective spaces to that of
weighted height on weighted projective spaces and show how such a height can be
computed. We prove some of the basic properties of the weighted height and show
how it can be used to study hyperelliptic curves over Q. Some examples are
provided from the weighted moduli space of binary sextics and octavics
Effective results for hyper- and superelliptic equations over number fields
We consider hyper- and superelliptic equations with unknowns x,y
from the ring of S-integers of a given number field K. Here, f is a polynomial
with S-integral coefficients of degree n with non-zero discriminant and b is a
non-zero S-integer. Assuming that n>2 if m=2 or n>1 if m>2, we give completely
explicit upper bounds for the heights of the solutions x,y in terms of the
heights of f and b, the discriminant of K, and the norms of the prime ideals in
S. Further, we give a completely explicit bound C such that has no
solutions in S-integers x,y if m>C, except if y is 0 or a root of unity. We
will apply these results in another paper where we consider hyper- and
superelliptic equations with unknowns taken from an arbitrary finitely
generated integral domain of characteristic 0.Comment: 31 page
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