147,870 research outputs found
1-Factors and Polynomials
In this paper we give a fuller exposition of a property of 1-factors discussed in [1]. The 1-factors of cubic graphs are found to be enumerated by a graph-function closley related to the chromatic and flow polynomials. The first part of the paper is a short account, with some minor improvements, of the theory of V-functions and φ-functions first set out in [1]
Factoring bivariate lacunary polynomials without heights
We present an algorithm which computes the multilinear factors of bivariate
lacunary polynomials. It is based on a new Gap Theorem which allows to test
whether a polynomial of the form P(X,X+1) is identically zero in time
polynomial in the number of terms of P(X,Y). The algorithm we obtain is more
elementary than the one by Kaltofen and Koiran (ISSAC'05) since it relies on
the valuation of polynomials of the previous form instead of the height of the
coefficients. As a result, it can be used to find some linear factors of
bivariate lacunary polynomials over a field of large finite characteristic in
probabilistic polynomial time.Comment: 25 pages, 1 appendi
Mellin transforms with only critical zeros: generalized Hermite functions
We consider the Mellin transforms of certain generalized Hermite functions
based upon certain generalized Hermite polynomials, characterized by a
parameter . We show that the transforms have polynomial factors whose
zeros lie all on the critical line. The polynomials with zeros only on the
critical line are identified in terms of certain hypergeometric
functions, being certain scaled and shifted Meixner-Pollaczek polynomials.
Other results of special function theory are presented.Comment: 17 pages, no figure
A note on quantum algorithms and the minimal degree of epsilon-error polynomials for symmetric functions
The degrees of polynomials representing or approximating Boolean functions
are a prominent tool in various branches of complexity theory. Sherstov
recently characterized the minimal degree deg_{\eps}(f) among all polynomials
(over the reals) that approximate a symmetric function f:{0,1}^n-->{0,1} up to
worst-case error \eps: deg_{\eps}(f) = ~\Theta(deg_{1/3}(f) +
\sqrt{n\log(1/\eps)}). In this note we show how a tighter version (without the
log-factors hidden in the ~\Theta-notation), can be derived quite easily using
the close connection between polynomials and quantum algorithms.Comment: 7 pages LaTeX. 2nd version: corrected a few small inaccuracie
Ergodic averages of commuting transformations with distinct degree polynomial iterates
We prove mean convergence, as , for the multiple ergodic averages
,
where are integer polynomials with distinct degrees, and
are commuting, invertible measure preserving transformations,
acting on the same probability space. This establishes several cases of a
conjecture of Bergelson and Leibman, that complement the case of linear
polynomials, recently established by Tao. Furthermore, we show that, unlike the
case of linear polynomials, for polynomials of distinct degrees, the
corresponding characteristic factors are mixtures of inverse limits of
nilsystems. We use this particular structure, together with some
equidistribution results on nilmanifolds, to give an application to multiple
recurrence and a corresponding one to combinatorics.Comment: 44 pages, small correction in the proof of Lemma 7.5, appeared in the
Proceedings of the London Mathematical Societ
- …