arXiv subjects

Wing-Keung To

Publications and source records attributed to Wing-Keung To.

5 recordsLinked to original sources

arXiv2026

Positivity of polynomials on the nonnegative part of certain affine hypersurfaces

We consider polynomials on the intersection of the closed positive orthant with the height-$1$ level hypersurface of certain polynomials with positive coefficients. We show that any polynomial strictly positive on such a semi-algebraic set can be represented by some polynomial with only positive coefficients. This result generalizes a result of P\'olya which corresponds to the case when the semi-algebraic set is the standard simplex. Our proof uses the Archimedean Representation Theorem from real algebra.

math.AG↗

arXiv2019

Characterization of polynomials whose large powers have fully positive coefficients

We give a criterion which characterizes a real multi-variate Laurent polynomial with full-dimensional smooth Newton polytope to have the property that all sufficiently large powers of the polynomial have fully positive coefficients. Here a Laurent polynomial is said to have fully positive coefficients if the coefficients of its monomial terms indexed by the lattice points of its Newton polytope are all positive. Our result generalizes an earlier result of the authors, which corresponds to the special case when the Newton polytope of the Laurent polynomial is a translate of a standard simplex. The result also generalizes a result of De Angelis, which corresponds to the special case of univariate polynomials. As an application, we also give a characterization of certain polynomial spectral radius functions of the defining matrix functions of Markov chains.

math.AG↗

arXiv2017

Characterization of polynomials whose large powers have all positive coefficients

We give a criterion which characterizes a homogeneous real multi-variate polynomial to have the property that all sufficiently large powers of the polynomial (as well as their products with any given positive homogeneous polynomial) have positive coefficients. Our result generalizes a result of De Angelis, which corresponds to the case of homogeneous bi-variate polynomials, as well as a classical result of Pólya, which corresponds to the case of a specific linear polynomial. As an application, we also give a characterization of certain polynomial beta functions, which are the spectral radius functions of the defining matrix functions of Markov chains.

math.CV↗

arXiv2016

Eventual positivity of Hermitian algebraic functions and associated integral operators

Quillen proved that repeated multiplication of the standard sesquilinear form to a positive Hermitian bihomogeneous polynomial eventually results in a sum of Hermitian squares, which was the first Hermitian analogue of Hilbert's seventeenth problem in the nondegenerate case. Later Catlin-D'Angelo generalized this positivstellensatz of Quillen to the case of Hermitian algebraic functions on holomorphic line bundles over compact complex manifolds by proving the eventual positivity of an associated integral operator. The arguments of Catlin-D'Angelo, as well as that of a subsequent refinement by Varolin, involve subtle asymptotic estimates of the Bergman kernel. In this article, we give an elementary and geometric proof of the eventual positivity of this integral operator, thereby yielding another proof of the corresponding positivstellensatz.

math.DG↗

arXiv2010

Buser-Sarnak invariant and projective normality of abelian varieties

We show that a general $n$-dimensional polarized abelian variety $(A,L)$ of a given polarization type and satisfying $ h^0(A, L) \geq \dfrac{8^n}{2} \cdot \dfrac{n^n}{n !}$ is projectively normal. In the process, we also obtain a sharp lower bound for the volume of a purely one-dimensional complex analytic subvariety in a geodesic tubular neighborhood of a subtorus of a compact complex torus.

math.AG↗