54,123 research outputs found
Algebraic arctic curves in the domain-wall six-vertex model
The arctic curve, i.e. the spatial curve separating ordered (or `frozen') and
disordered (or `temperate) regions, of the six-vertex model with domain wall
boundary conditions is discussed for the root-of-unity vertex weights. In these
cases the curve is described by algebraic equations which can be worked out
explicitly from the parametric solution for this curve. Some interesting
examples are discussed in detail. The upper bound on the maximal degree of the
equation in a generic root-of-unity case is obtained.Comment: 15 pages, no figures; v2: metadata correcte
Root separation for irreducible integer polynomials
We establish new results on root separation of integer, irreducible
polynomials of degree at least four. These improve earlier bounds of Bugeaud
and Mignotte (for even degree) and of Beresnevich, Bernik, and Goetze (for odd
degree).Comment: 8 pages; revised version; to appear in Bull. Lond. Math. So
Root separation for reducible monic polynomials of odd degree
We study root separation of reducible monic integer polynomials of odd
degree. Let h(P) be the naive height, sep(P) the minimal distance between two
distinct roots of an integer polynomial P(x) and sep(P)=h(P)^{-e(P)}. Let
e_r*(d)=limsup_{deg(P)=d, h(P)-> +infty} e(P), where the limsup is taken over
the reducible monic integer polynomials P(x) of degree d. We prove that e_r*(d)
<= d-2. We also obtain a lower bound for e_r*(d) for d odd, which improves
previously known lower bounds for e_r*(d) when d = 5, 7, 9.Comment: 8 pages, to appear in Rad Hrvat. Akad. Znan. Umjet. Mat. Zna
On bi-integrable natural Hamiltonian systems on the Riemannian manifolds
We introduce the concept of natural Poisson bivectors, which generalizes the
Benenti approach to construction of natural integrable systems on the
Riemannian manifolds and allows us to consider almost the whole known zoo of
integrable systems in framework of bi-hamiltonian geometry.Comment: 24 pages, LaTeX with AMSfonts (some new references were added
Log-majorization of the moduli of the eigenvalues of a matrix polynomial by tropical roots
We show that the sequence of moduli of the eigenvalues of a matrix polynomial
is log-majorized, up to universal constants, by a sequence of "tropical roots"
depending only on the norms of the matrix coefficients. These tropical roots
are the non-differentiability points of an auxiliary tropical polynomial, or
equivalently, the opposites of the slopes of its Newton polygon. This extends
to the case of matrix polynomials some bounds obtained by Hadamard, Ostrowski
and P\'olya for the roots of scalar polynomials. We also obtain new bounds in
the scalar case, which are accurate for "fewnomials" or when the tropical roots
are well separated.Comment: 36 pages, 19 figure
A Generic Position Based Method for Real Root Isolation of Zero-Dimensional Polynomial Systems
We improve the local generic position method for isolating the real roots of
a zero-dimensional bivariate polynomial system with two polynomials and extend
the method to general zero-dimensional polynomial systems. The method mainly
involves resultant computation and real root isolation of univariate polynomial
equations. The roots of the system have a linear univariate representation. The
complexity of the method is for the bivariate case, where
, resp., is an upper bound on the degree, resp., the
maximal coefficient bitsize of the input polynomials. The algorithm is
certified with probability 1 in the multivariate case. The implementation shows
that the method is efficient, especially for bivariate polynomial systems.Comment: 24 pages, 5 figure
Symbolic-Numeric Tools for Analytic Combinatorics in Several Variables
Analytic combinatorics studies the asymptotic behaviour of sequences through
the analytic properties of their generating functions. This article provides
effective algorithms required for the study of analytic combinatorics in
several variables, together with their complexity analyses. Given a
multivariate rational function we show how to compute its smooth isolated
critical points, with respect to a polynomial map encoding asymptotic
behaviour, in complexity singly exponential in the degree of its denominator.
We introduce a numerical Kronecker representation for solutions of polynomial
systems with rational coefficients and show that it can be used to decide
several properties (0 coordinate, equal coordinates, sign conditions for real
solutions, and vanishing of a polynomial) in good bit complexity. Among the
critical points, those that are minimal---a property governed by inequalities
on the moduli of the coordinates---typically determine the dominant asymptotics
of the diagonal coefficient sequence. When the Taylor expansion at the origin
has all non-negative coefficients (known as the `combinatorial case') and under
regularity conditions, we utilize this Kronecker representation to determine
probabilistically the minimal critical points in complexity singly exponential
in the degree of the denominator, with good control over the exponent in the
bit complexity estimate. Generically in the combinatorial case, this allows one
to automatically and rigorously determine asymptotics for the diagonal
coefficient sequence. Examples obtained with a preliminary implementation show
the wide applicability of this approach.Comment: As accepted to proceedings of ISSAC 201
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