251 research outputs found
Effective results for discriminant equations over finitely generated domains
Let be an integral domain with quotient field of characteristic
that is finitely generated as a -algebra. Denote by the
discriminant of a polynomial . Further, given a finite etale algebra
, we denote by the discriminant of
over . For non-zero , we consider equations
to be solved in monic polynomials of given degree having
their zeros in a given finite extension field of , and
D_{\Omega/K}(\alpha)=\delta\,\,\mbox{ in } \alpha\in O, where is an
-order of , i.e., a subring of the integral closure of in
that contains as well as a -basis of .
In our book ``Discriminant Equations in Diophantine Number Theory, which will
be published by Cambridge University Press we proved that if is effectively
given in a well-defined sense and integrally closed, then up to natural notions
of equivalence the above equations have only finitely many solutions, and that
moreover, a full system of representatives for the equivalence classes can be
determined effectively. In the present paper, we extend these results to
integral domains that are not necessarily integrally closed.Comment: 20 page
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
Effective results for Diophantine equations over finitely generated domains
Let A be an arbitrary integral domain of characteristic 0 which is finitely
generated over Z. We consider Thue equations with unknowns x,y from
A and hyper- and superelliptic equations with unknowns from A,
where the binary form F and the polynomial f have their coefficients in A,
where b is a non-zero element from A, and where m is an integer at least 2.
Under the necessary finiteness conditions imposed on F,f,m, we give explicit
upper bounds for the sizes of x,y in terms of suitable representations for
A,F,f,b Our results imply that the solutions of Thue equations and hyper- and
superelliptic equations over arbitrary finitely generated domains can be
determined effectively in principle. Further, we generalize a theorem of
Schinzel and Tijdeman to the effect, that there is an effectively computable
constant C such that has no solutions in x,y from A with y not 0 or
a root of unity if m>C. In our proofs, we use effective results for Thue
equations and hyper- and superelliptic equations over number fields and
function fields, some effective commutative algebra, and a specialization
argument.Comment: 37 page
On the denominators of equivalent algebraic numbers
This article does not have an abstract
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