23,112 research outputs found
Development of symbolic algorithms for certain algebraic processes
This study investigates the problem of computing the exact greatest common divisor of two polynomials relative to an orthogonal basis, defined over the rational number field. The main objective of the study is to design and implement an effective and efficient symbolic algorithm for the general class of dense polynomials, given the rational number defining terms of their basis. From a general algorithm using the comrade matrix approach, the nonmodular and modular techniques are prescribed. If the coefficients of the generalized polynomials are multiprecision integers, multiprecision arithmetic will be required in the construction of the comrade matrix and the corresponding systems coefficient matrix. In addition, the application of the nonmodular elimination technique on this coefficient matrix extensively applies multiprecision rational number operations. The modular technique is employed to minimize the complexity involved in such computations. A divisor test algorithm that enables the detection of an unlucky reduction is a crucial device for an effective implementation of the modular technique. With the bound of the true solution not known a priori, the test is devised and carefully incorporated into the modular algorithm. The results illustrate that the modular algorithm illustrate its best performance for the class of relatively prime polynomials. The empirical computing time results show that the modular algorithm is markedly superior to the nonmodular algorithms in the case of sufficiently dense Legendre basis polynomials with a small GCD solution. In the case of dense Legendre basis polynomials with a big GCD solution, the modular algorithm is significantly superior to the nonmodular algorithms in higher degree polynomials. For more definitive conclusions, the computing time functions of the algorithms that are presented in this report have been worked out. Further investigations have also been suggested
Resolving zero-divisors using Hensel lifting
Algorithms which compute modulo triangular sets must respect the presence of
zero-divisors. We present Hensel lifting as a tool for dealing with them. We
give an application: a modular algorithm for computing GCDs of univariate
polynomials with coefficients modulo a radical triangular set over the
rationals. Our modular algorithm naturally generalizes previous work from
algebraic number theory. We have implemented our algorithm using Maple's RECDEN
package. We compare our implementation with the procedure RegularGcd in the
RegularChains package.Comment: Shorter version to appear in Proceedings of SYNASC 201
The complexity of class polynomial computation via floating point approximations
We analyse the complexity of computing class polynomials, that are an
important ingredient for CM constructions of elliptic curves, via complex
floating point approximations of their roots. The heart of the algorithm is the
evaluation of modular functions in several arguments. The fastest one of the
presented approaches uses a technique devised by Dupont to evaluate modular
functions by Newton iterations on an expression involving the
arithmetic-geometric mean. It runs in time for any , where
is the CM discriminant and is the degree of the class polynomial.
Another fast algorithm uses multipoint evaluation techniques known from
symbolic computation; its asymptotic complexity is worse by a factor of . Up to logarithmic factors, this running time matches the size of the
constructed polynomials. The estimate also relies on a new result concerning
the complexity of enumerating the class group of an imaginary-quadratic order
and on a rigorously proven upper bound for the height of class polynomials
Fast arithmetic in unramified p-adic fields
Let p be prime and Zpn the degree n unramified extension of the ring of
p-adic integers Zp. In this paper we give an overview of some very fast
algorithms for common operations in Zpn modulo p^N. Combining existing methods
with recent work of Kedlaya and Umans about modular composition of polynomials,
we achieve quasi-linear time algorithms in the parameters n and N, and
quasi-linear or quasi-quadratic time in log p, for most basic operations on
these fields, including Galois conjugation, Teichmuller lifting and computing
minimal polynomials.Comment: 6 page
Topological string amplitudes for the local half K3 surface
We study topological string amplitudes for the local half K3 surface. We
develop a method of computing higher-genus amplitudes along the lines of the
direct integration formalism, making full use of the Seiberg-Witten curve
expressed in terms of modular forms and E_8-invariant Jacobi forms. The
Seiberg-Witten curve was constructed previously for the low-energy effective
theory of the non-critical E-string theory in R^4 x T^2. We clarify how the
amplitudes are written as polynomials in a finite number of generators
expressed in terms of the Seiberg-Witten curve. We determine the coefficients
of the polynomials by solving the holomorphic anomaly equation and the gap
condition, and construct the amplitudes explicitly up to genus three. The
results encompass topological string amplitudes for all local del Pezzo
surfaces.Comment: 35 pages, v2: several clarifications made, an equation and references
added, v3: published versio
Explicit CM-theory for level 2-structures on abelian surfaces
For a complex abelian variety with endomorphism ring isomorphic to the
maximal order in a quartic CM-field , the Igusa invariants generate an abelian extension of the reflex field of . In
this paper we give an explicit description of the Galois action of the class
group of this reflex field on . We give a geometric
description which can be expressed by maps between various Siegel modular
varieties. We can explicitly compute this action for ideals of small norm, and
this allows us to improve the CRT method for computing Igusa class polynomials.
Furthermore, we find cycles in isogeny graphs for abelian surfaces, thereby
implying that the `isogeny volcano' algorithm to compute endomorphism rings of
ordinary elliptic curves over finite fields does not have a straightforward
generalization to computing endomorphism rings of abelian surfaces over finite
fields
Modular polynomials via isogeny volcanoes
We present a new algorithm to compute the classical modular polynomial Phi_n
in the rings Z[X,Y] and (Z/mZ)[X,Y], for a prime n and any positive integer m.
Our approach uses the graph of n-isogenies to efficiently compute Phi_n mod p
for many primes p of a suitable form, and then applies the Chinese Remainder
Theorem (CRT). Under the Generalized Riemann Hypothesis (GRH), we achieve an
expected running time of O(n^3 (log n)^3 log log n), and compute Phi_n mod m
using O(n^2 (log n)^2 + n^2 log m) space. We have used the new algorithm to
compute Phi_n with n over 5000, and Phi_n mod m with n over 20000. We also
consider several modular functions g for which Phi_n^g is smaller than Phi_n,
allowing us to handle n over 60000.Comment: corrected a typo in equation (14), 31 page
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