47 research outputs found
A unified approach to polynomial sequences with only real zeros
We give new sufficient conditions for a sequence of polynomials to have only
real zeros based on the method of interlacing zeros. As applications we derive
several well-known facts, including the reality of zeros of orthogonal
polynomials, matching polynomials, Narayana polynomials and Eulerian
polynomials. We also settle certain conjectures of Stahl on genus polynomials
by proving them for certain classes of graphs, while showing that they are
false in general.Comment: 19 pages, Advances in Applied Mathematics, in pres
Jensen polynomials for the Riemann zeta function and other sequences
In 1927 P\'olya proved that the Riemann Hypothesis is equivalent to the
hyperbolicity of Jensen polynomials for the Riemann zeta function at
its point of symmetry. This hyperbolicity has been proved for degrees . We obtain an asymptotic formula for the central derivatives
that is accurate to all orders, which allows us to prove
the hyperbolicity of a density subset of the Jensen polynomials of each
degree. Moreover, we establish hyperbolicity for all . These results
follow from a general theorem which models such polynomials by Hermite
polynomials. In the case of the Riemann zeta function, this proves the GUE
random matrix model prediction in derivative aspect. The general theorem also
allows us to prove a conjecture of Chen, Jia, and Wang on the partition
function.Comment: 11 page
Jensen polynomials for the Riemann zeta function and other sequences
In 1927 PĂłlya proved that the Riemann Hypothesis is equivalent to the hyperbolicity of Jensen polynomials for the Riemann zeta function at its point of symmetry. This hyperbolicity has been proved for degrees . We obtain an asymptotic formula for the central derivatives that is accurate to all orders, which allows us to prove the hyperbolicity of a density subset of the Jensen polynomials of each degree. Moreover, we establish hyperbolicity for all . These results follow from a general theorem which models such polynomials by Hermite polynomials. In the case of the Riemann zeta function, this proves the Gaussian unitary ensemble random matrix model prediction in derivative aspect. The general theorem also allows us to prove a conjecture of Chen, Jia, and Wang on the partition function
Combinatorial and Additive Number Theory Problem Sessions: '09--'19
These notes are a summary of the problem session discussions at various CANT
(Combinatorial and Additive Number Theory Conferences). Currently they include
all years from 2009 through 2019 (inclusive); the goal is to supplement this
file each year. These additions will include the problem session notes from
that year, and occasionally discussions on progress on previous problems. If
you are interested in pursuing any of these problems and want additional
information as to progress, please email the author. See
http://www.theoryofnumbers.com/ for the conference homepage.Comment: Version 3.4, 58 pages, 2 figures added 2019 problems on 5/31/2019,
fixed a few issues from some presenters 6/29/201
Geometric and Topological Combinatorics
The 2007 Oberwolfach meeting “Geometric and Topological Combinatorics” presented a great variety of investigations where topological and algebraic methods are brought into play to solve combinatorial and geometric problems, but also where geometric and combinatorial ideas are applied to topological questions
Permutation classes
This is a survey on permutation classes for the upcoming book Handbook of
Enumerative Combinatorics