862 research outputs found
Modular Las Vegas Algorithms for Polynomial Absolute Factorization
Let f(X,Y) \in \ZZ[X,Y] be an irreducible polynomial over \QQ. We give a
Las Vegas absolute irreducibility test based on a property of the Newton
polytope of , or more precisely, of modulo some prime integer . The
same idea of choosing a satisfying some prescribed properties together with
is used to provide a new strategy for absolute factorization of .
We present our approach in the bivariate case but the techniques extend to the
multivariate case. Maple computations show that it is efficient and promising
as we are able to factorize some polynomials of degree up to 400
Subgeometric ergodicity of strong Markov processes
We derive sufficient conditions for subgeometric f-ergodicity of strongly
Markovian processes. We first propose a criterion based on modulated moment of
some delayed return-time to a petite set. We then formulate a criterion for
polynomial f-ergodicity in terms of a drift condition on the generator.
Applications to specific processes are considered, including Langevin tempered
diffusions on R^n and storage models.Comment: Published at http://dx.doi.org/10.1214/105051605000000115 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
A Multi-level Blocking Distinct Degree Factorization Algorithm
We give a new algorithm for performing the distinct-degree factorization of a
polynomial P(x) over GF(2), using a multi-level blocking strategy. The coarsest
level of blocking replaces GCD computations by multiplications, as suggested by
Pollard (1975), von zur Gathen and Shoup (1992), and others. The novelty of our
approach is that a finer level of blocking replaces multiplications by
squarings, which speeds up the computation in GF(2)[x]/P(x) of certain interval
polynomials when P(x) is sparse. As an application we give a fast algorithm to
search for all irreducible trinomials x^r + x^s + 1 of degree r over GF(2),
while producing a certificate that can be checked in less time than the full
search. Naive algorithms cost O(r^2) per trinomial, thus O(r^3) to search over
all trinomials of given degree r. Under a plausible assumption about the
distribution of factors of trinomials, the new algorithm has complexity O(r^2
(log r)^{3/2}(log log r)^{1/2}) for the search over all trinomials of degree r.
Our implementation achieves a speedup of greater than a factor of 560 over the
naive algorithm in the case r = 24036583 (a Mersenne exponent). Using our
program, we have found two new primitive trinomials of degree 24036583 over
GF(2) (the previous record degree was 6972593)
The Painlev\'e methods
This short review is an introduction to a great variety of methods, the
collection of which is called the Painlev\'e analysis, intended at producing
all kinds of exact (as opposed to perturbative) results on nonlinear equations,
whether ordinary, partial, or discrete.Comment: LaTex 2e, subject index, Nonlinear integrable systems: classical and
quantum, ed. A. Kundu, Special issue, Proceedings of Indian Science Academy,
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