Results 1  10
of
1,913
Factoring polynomials with rational coefficients
 MATH. ANN
, 1982
"... In this paper we present a polynomialtime algorithm to solve the following problem: given a nonzero polynomial fe Q[X] in one variable with rational coefficients, find the decomposition of f into irreducible factors in Q[X]. It is well known that this is equivalent to factoring primitive polynomia ..."
Abstract

Cited by 961 (11 self)
 Add to MetaCart
In this paper we present a polynomialtime algorithm to solve the following problem: given a nonzero polynomial fe Q[X] in one variable with rational coefficients, find the decomposition of f into irreducible factors in Q[X]. It is well known that this is equivalent to factoring primitive
Asymptotics of Linear Recurrences with Rational Coefficients
 FORMAL POWER SERIES AND ALGEBRAIC COMBINATORICS
, 1993
"... We give algorithms to compute the asymptotic expansion of solutions of linear recurrences with rational coefficients and rational initial conditions in polynomial time in the order of the recurrence. ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
We give algorithms to compute the asymptotic expansion of solutions of linear recurrences with rational coefficients and rational initial conditions in polynomial time in the order of the recurrence.
CM newforms with rational coefficients

, 2008
"... We classify newforms with rational Fourier coefficients and complex multiplication for fixed weight up to twisting. Under the extended Riemann hypothesis for odd real Dirichlet characters, these newforms are finite in number. We produce tables for weights 3 and 4, where finiteness holds unconditiona ..."
Abstract

Cited by 17 (6 self)
 Add to MetaCart
We classify newforms with rational Fourier coefficients and complex multiplication for fixed weight up to twisting. Under the extended Riemann hypothesis for odd real Dirichlet characters, these newforms are finite in number. We produce tables for weights 3 and 4, where finiteness holds
Factoring Polynomials with Rational Coefficients
, 1998
"... this paper is to present, in greater detail, this algorithm for factoring polynomials over Q. It is based on the original paper of Lenstra, Lenstra, and Lov'asz [4] of the same title and on courses in algebraic number theory and the geometry of numbers, taught by Cameron Stewart at the Universi ..."
Abstract
 Add to MetaCart
at the University of Waterloo. 2 Motivation For notational purposes, for any f ffl Z[x], let (f mod m) denote the polynomial in Z=mZ[x]whose coefficients are the respective coefficients of f reduced modulo m. In order for the algorithm to run correctly, we must choose choose our values of p and k very carefully so
Factorization of Univariate Polynomials with Rational Coefficients
"... Let g1,...,g k be the true factors of f in Z[x] and let f1,...,fr be the local factors (over the padic integers). Current implementations Hensel lift to determine f1,...,fr with a padic accuracy a that is guaranteed to be high enough to recover any potential factor of f in Z[x]. However, the probl ..."
Abstract
 Add to MetaCart
small factors, say g2,...,g k. Then, to recover g1,...,g k we do not need p a to be larger than twice the largest coefficient of g1. All we need is that p a is larger than twice the largest coefficient in g2,...,g k. This suffices to reconstruct g2,...,g k ∈ Z[x] from their modular images, after which
Effective Asymptotics of Linear Recurrences with Rational Coefficients
 Discrete Mathematics
, 1996
"... We give algorithms to compute the asymptotic expansion of solutions of linear recurrences with rational coefficients and rational initial conditions in polynomial time in the order of the recurrence. Introduction We investigate sequences defined by a recurrence of the form a k un+k + a k\Gamma1 un+k ..."
Abstract

Cited by 10 (1 self)
 Add to MetaCart
We give algorithms to compute the asymptotic expansion of solutions of linear recurrences with rational coefficients and rational initial conditions in polynomial time in the order of the recurrence. Introduction We investigate sequences defined by a recurrence of the form a k un+k + a k\Gamma1 un
Transcendence of Formal Power Series With Rational Coefficients
, 1999
"... We give algebraic proofs of transcendence over Q(X) of formal power series with rational coefficients, by using inter alia reduction modulo prime numbers, and the Christol theorem. Applications to generating series of languages and combinatorial objects are given. Keywords: transcendental formal po ..."
Abstract

Cited by 12 (3 self)
 Add to MetaCart
We give algebraic proofs of transcendence over Q(X) of formal power series with rational coefficients, by using inter alia reduction modulo prime numbers, and the Christol theorem. Applications to generating series of languages and combinatorial objects are given. Keywords: transcendental formal
Hecke eigenforms with rational coefficients and complex multiplication
, 2008
"... We prove that, assuming GRH, there are only finitely many newforms with rational Fourier coefficients and complex multiplication for fixed weight up to twisting. In the sequel, we produce tables of such forms for weights 3 and 4, where this holds unconditionally. We also comment on geometric realiza ..."
Abstract

Cited by 15 (10 self)
 Add to MetaCart
We prove that, assuming GRH, there are only finitely many newforms with rational Fourier coefficients and complex multiplication for fixed weight up to twisting. In the sequel, we produce tables of such forms for weights 3 and 4, where this holds unconditionally. We also comment on geometric
ON THE ZEROS OF LINEAR DIFFERENTIAL POLYNOMIALS WITH SMALL RATIONAL COEFFICIENTS
"... We prove the following: suppose that J{z) is transcendental and meromorphic of finite order in the plane, and that the linear differential polynomial F(z) is defined by and is nonconstant, where ak_j(z),...,ao(z) are rational functions vanishing at infinity. Then implies that N(r,l/(fFF')) = N ..."
Abstract
 Add to MetaCart
We prove the following: suppose that J{z) is transcendental and meromorphic of finite order in the plane, and that the linear differential polynomial F(z) is defined by and is nonconstant, where ak_j(z),...,ao(z) are rational functions vanishing at infinity. Then implies that N
Results 1  10
of
1,913