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Effective results for hyper- and superelliptic equations over number fields | Attila Bérczes
; Jan-Hendrik Evertse
; Kálmán Györy
; | Date: |
30 Jan 2013 | Abstract: | We consider hyper- and superelliptic equations $f(x)=by^m$ 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 $f(x)=by^m$ 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. | Source: | arXiv, 1301.7168 | Services: | Forum | Review | PDF | Favorites |
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