12,944 research outputs found
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
Root numbers and the parity problem
Let E be a one-parameter family of elliptic curves over a number field. It is
natural to expect the average root number of the curves in the family to be
zero. All known counterexamples to this folk conjecture occur for families
obeying a certain degeneracy condition. We prove that the average root number
is zero for a large class of families of elliptic curves of fairly general
type. Furthermore, we show that any non-degenerate family E has average root
number 0, provided that two classical arithmetical conjectures hold for two
homogeneous polynomials with integral coefficients constructed explicitly in
terms of E.
The first such conjecture -- commonly associated with Chowla -- asserts the
equidistribution of the parity of the number of primes dividing the integers
represented by a polynomial. We prove the conjecture for homogeneous
polynomials of degree 3.
The second conjecture used states that any non-constant homogeneous
polynomial yields to a square-free sieve. We sharpen the existing bounds on the
known cases by a sieve refinement and a new approach combining height
functions, sphere packings and sieve methods.Comment: 291 pages, PhD thesi
Forecasting time series with sieve bootstrap
In this paper we consider bootstrap methods for constructing nonparametric prediction intervals for a general class of linear processes. Our approach uses the sieve bootstrap procedure of Biihlmann (1997) based on residual resampling from an autoregressive approximation to the given process. We show that the sieve bootstrap provides consistent estimators of the conditional distribution of future values given the observed data, assuming that the order of the autoregressive approximation increases with the sample size at a suitable rate and some restrictions about polynomial decay of the coefficients ~ j t:o of the process MA(oo) representation. We present a Monte Carlo study comparing the finite sample properties of the sieve bootstrap with those of alternative methods. Finally, we illustrate the performance of the proposed method with real data examples
Root optimization of polynomials in the number field sieve
The general number field sieve (GNFS) is the most efficient algorithm known
for factoring large integers. It consists of several stages, the first one
being polynomial selection. The quality of the chosen polynomials in polynomial
selection can be modelled in terms of size and root properties. In this paper,
we describe some algorithms for selecting polynomials with very good root
properties.Comment: 16 pages, 18 reference
The inverse sieve problem in high dimensions
We show that if a big set of integer points in [0,N]^d, d>1, occupies few
residue classes mod p for many primes p, then it must essentially lie in the
solution set of some polynomial equation of low degree. This answers a question
of Helfgott and Venkatesh.Comment: 15 pages. Added more examples in Section 5 and some minor change
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