1,547 research outputs found
An elementary approach to polynomial optimization on polynomial meshes
A polynomial mesh on a multivariate compact set or manifold is a sequence of finite norming sets for polynomials whose norming constant is independent of degree. We apply the recently developed theory of polynomial meshes to an elementary discrete approach for polynomial optimization on nonstandard domains, providing a rigorous (over)estimate of the convergence rate. Examples include surface/solid subregions of sphere or torus, such as caps, lenses, lunes, and slices
Oscillation of linear ordinary differential equations: on a theorem by A. Grigoriev
We give a simplified proof and an improvement of a recent theorem by A.
Grigoriev, placing an upper bound for the number of roots of linear
combinations of solutions to systems of linear equations with polynomial or
rational coefficients.Comment: 16 page
Redundant Picard-Fuchs system for Abelian integrals
We derive an explicit system of Picard-Fuchs differential equations satisfied
by Abelian integrals of monomial forms and majorize its coefficients. A
peculiar feature of this construction is that the system admitting such
explicit majorants, appears only in dimension approximately two times greater
than the standard Picard-Fuchs system.
The result is used to obtain a partial solution to the tangential Hilbert
16th problem. We establish upper bounds for the number of zeros of arbitrary
Abelian integrals on a positive distance from the critical locus. Under the
additional assumption that the critical values of the Hamiltonian are distant
from each other (after a proper normalization), we were able to majorize the
number of all (real and complex) zeros.
In the second part of the paper an equivariant formulation of the above
problem is discussed and relationships between spread of critical values and
non-homogeneity of uni- and bivariate complex polynomials are studied.Comment: 31 page, LaTeX2e (amsart
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