6 research outputs found
A new notion of majorization for polynomials
In this paper, we introduce a notion called strong majorization for
realrooted polynomials, and we show how it relates to standard majorization and
how it can be checked through a simple fraction decomposition
A theory of singular values for finite free probability
We introduce a finite version of free probability for rectangular matrices
that amounts to operations on singular values of polynomials. We show that we
can replicate the transforms from free probability, and that asymptotically
there is convergence from rectangular finite free probability to rectangular
free probability. Lastly, we show that classical distribution results such as a
law of large numbers or a central limit theorem can be made explicit in this
new framework where random variables are replaced by polynomials
On a new simple discriminant inequality
We prove a simple though nontrivial inequality involving a real-rooted
polynomial, its successive derivatives and their discriminants. In particular
we get the new inequality : \Dis(p-tp')>\Dis(p) for a polynomial , its
derivative and t any non zero real. We also formulate a well-motivated
conjecture generalizing our main result
A rectangular additive convolution for polynomials
We define the rectangular additive convolution of polynomials with
nonnegative real roots as a generalization of the asymmetric additive
convolution introduced by Marcus, Spielman and Srivastava. We then prove a
sliding bound on the largest root of this convolution. The main tool used in
the analysis is a differential operator derived from the "rectangular Cauchy
transform" introduced by Benaych-Georges. The proof is inductive, with the base
case requiring a new nonasymptotic bound on the Cauchy transform of Gegenbauer
polynomials which may be of independent interest
Special polynomials and new real-rootedness results
In this paper, we exhibit new monotonicity properties of roots of families of
orthogonal polynomials depending polynomially on a parameter
(Laguerre and Gegenbauer). We show that are realrooted in
for in the support of orthogonality. As an application we show
realrootedness in and interlacing properties of
for and , establishing a dual approach to orthogonality
Rectangular finite free probability theory
We study a new type of polynomial convolution that serves as the foundation for building what we call rectangular finite free probability theory, generalizing the square finite free probability theory of Marcus, Spielman and Srivastava. We relate this operation to large rectangular random matrices and explain how it acts on singular values of rectangular matrices in a canonical way. Furthermore, we obtain nontrivial inequalities on roots of polynomials and build some appropriate tools, e.g. the analogue of the classical -transforms. These developments are inspired by well-known results and concepts from probability theory. We also show that classical orthogonal polynomials such as Gegenbauer or Laguerre polynomials naturally arise through this convolution. Consequently, we deduce new nontrivial properties about the positions of the roots of these polynomials. As an application, we give an elegant proof of the existence of biregular bipartite Ramanujan graphs