872 research outputs found
Equations in the Hadamard ring of rational functions
Let k be a number field. It is well known that the set of sequences composed
by Taylor coefficients of rational functions over k is closed under
component-wise operations, and so it can be equipped with a ring structure. A
conjecture due to Pisot asks if (after enlarging the field) one can take d-th
roots in this ring, provided d-th roots of coefficients can be taken in k. This
was proved true in a preceding paper of the second author; in this article we
generalize this result to more general equations, monic in Y, where the former
case can be recovered for g(X,Y)=X^d-Y=0. Combining this with the Hadamard
quotient theorem by Pourchet and Van der Poorten, we are able to get rid of the
monic restriction, and have a theorem that generalizes both results.Comment: 18 pages, LaTe
Algorithms on Ideal over Complex Multiplication order
We show in this paper that the Gentry-Szydlo algorithm for cyclotomic orders,
previously revisited by Lenstra-Silverberg, can be extended to
complex-multiplication (CM) orders, and even to a more general structure. This
algorithm allows to test equality over the polarized ideal class group, and
finds a generator of the polarized ideal in polynomial time. Also, the
algorithm allows to solve the norm equation over CM orders and the recent
reduction of principal ideals to the real suborder can also be performed in
polynomial time. Furthermore, we can also compute in polynomial time a unit of
an order of any number field given a (not very precise) approximation of it.
Our description of the Gentry-Szydlo algorithm is different from the original
and Lenstra- Silverberg's variant and we hope the simplifications made will
allow a deeper understanding. Finally, we show that the well-known speed-up for
enumeration and sieve algorithms for ideal lattices over power of two
cyclotomics can be generalized to any number field with many roots of unity.Comment: Full version of a paper submitted to ANT
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