23,471 research outputs found
Denominator Bounds and Polynomial Solutions for Systems of q-Recurrences over K(t) for Constant K
We consider systems A_\ell(t) y(q^\ell t) + ... + A_0(t) y(t) = b(t) of
higher order q-recurrence equations with rational coefficients. We extend a
method for finding a bound on the maximal power of t in the denominator of
arbitrary rational solutions y(t) as well as a method for bounding the degree
of polynomial solutions from the scalar case to the systems case. The approach
is direct and does not rely on uncoupling or reduction to a first order system.
Unlike in the scalar case this usually requires an initial transformation of
the system.Comment: 8 page
Design of quadrature rules for Müntz and Müntz-logarithmic polynomials using monomial transformation
A method for constructing the exact quadratures for Müntz and Müntz-logarithmic polynomials is presented. The algorithm does permit to anticipate the precision (machine precision) of the numerical integration of Müntz-logarithmic polynomials in terms of the number of Gauss-Legendre (GL) quadrature samples and monomial transformation order. To investigate in depth the properties of classical GL quadrature, we present new optimal asymptotic estimates for the remainder. In boundary element integrals this quadrature rule can be applied to evaluate singular functions with end-point singularity, singular kernel as well as smooth functions. The method is numerically stable, efficient, easy to be implemented. The rule has been fully tested and several numerical examples are included. The proposed quadrature method is more efficient in run-time evaluation than the existing methods for Müntz polynomial
Computation of Galois groups of rational polynomials
Computational Galois theory, in particular the problem of computing the
Galois group of a given polynomial is a very old problem. Currently, the best
algorithmic solution is Stauduhar's method. Computationally, one of the key
challenges in the application of Stauduhar's method is to find, for a given
pair of groups H<G a G-relative H-invariant, that is a multivariate polynomial
F that is H-invariant, but not G-invariant. While generic, theoretical methods
are known to find such F, in general they yield impractical answers. We give a
general method for computing invariants of large degree which improves on
previous known methods, as well as various special invariants that are derived
from the structure of the groups. We then apply our new invariants to the task
of computing the Galois groups of polynomials over the rational numbers,
resulting in the first practical degree independent algorithm.Comment: Improved version and new titl
Rational invariants of even ternary forms under the orthogonal group
In this article we determine a generating set of rational invariants of
minimal cardinality for the action of the orthogonal group on
the space of ternary forms of even degree . The
construction relies on two key ingredients: On one hand, the Slice Lemma allows
us to reduce the problem to dermining the invariants for the action on a
subspace of the finite subgroup of signed permutations. On the
other hand, our construction relies in a fundamental way on specific bases of
harmonic polynomials. These bases provide maps with prescribed
-equivariance properties. Our explicit construction of these
bases should be relevant well beyond the scope of this paper. The expression of
the -invariants can then be given in a compact form as the
composition of two equivariant maps. Instead of providing (cumbersome) explicit
expressions for the -invariants, we provide efficient algorithms
for their evaluation and rewriting. We also use the constructed
-invariants to determine the -orbit locus and
provide an algorithm for the inverse problem of finding an element in
with prescribed values for its invariants. These are
the computational issues relevant in brain imaging.Comment: v3 Changes: Reworked presentation of Neuroimaging application,
refinement of Definition 3.1. To appear in "Foundations of Computational
Mathematics
Erdos-Szekeres-type statements: Ramsey function and decidability in dimension 1
A classical and widely used lemma of Erdos and Szekeres asserts that for
every n there exists N such that every N-term sequence a of real numbers
contains an n-term increasing subsequence or an n-term nondecreasing
subsequence; quantitatively, the smallest N with this property equals
(n-1)^2+1. In the setting of the present paper, we express this lemma by saying
that the set of predicates Phi={x_1<x_2,x_1\ge x_2}$ is Erdos-Szekeres with
Ramsey function ES_Phi(n)=(n-1)^2+1.
In general, we consider an arbitrary finite set Phi={Phi_1,...,Phi_m} of
semialgebraic predicates, meaning that each Phi_j=Phi_j(x_1,...,x_k) is a
Boolean combination of polynomial equations and inequalities in some number k
of real variables. We define Phi to be Erdos-Szekeres if for every n there
exists N such that each N-term sequence a of real numbers has an n-term
subsequence b such that at least one of the Phi_j holds everywhere on b, which
means that Phi_j(b_{i_1},...,b_{i_k}) holds for every choice of indices
i_1,i_2,...,i_k, 1<=i_1<i_2<... <i_k<= n. We write ES_Phi(n) for the smallest N
with the above property.
We prove two main results. First, the Ramsey functions in this setting are at
most doubly exponential (and sometimes they are indeed doubly exponential): for
every Phi that is Erd\H{o}s--Szekeres, there is a constant C such that
ES_Phi(n) < exp(exp(Cn)). Second, there is an algorithm that, given Phi,
decides whether it is Erdos-Szekeres; thus, one-dimensional
Erdos-Szekeres-style theorems can in principle be proved automatically.Comment: minor fixes of the previous version. to appear in Duke Math.
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