21,065 research outputs found
On Asymptotics of Polynomial Eigenfunctions for Exactly-Solvable Differential Operators
In this paper we study the asymptotic zero distribution of eigenpolynomials
for degenerate exactly-solvable operators. We present an explicit conjecture
and partial results on the growth of the largest modulus of the roots of the
unique and monic n:th degree eigenpolynomial of any such operator as the degree
n tends to infinity. Based on this conjecture we deduce the algebraic equation
satified by the Cauchy transform of the asymptotic root measure of the properly
scaled eigenpolynomials, for which the union of all roots is conjecturally
contained in a compact set.Comment: 36 pages, 37 figures, to appear in Journal of Approximation Theor
Tropical bounds for eigenvalues of matrices
We show that for all k = 1,...,n the absolute value of the product of the k
largest eigenvalues of an n-by-n matrix A is bounded from above by the product
of the k largest tropical eigenvalues of the matrix |A| (entrywise absolute
value), up to a combinatorial constant depending only on k and on the pattern
of the matrix. This generalizes an inequality by Friedland (1986),
corresponding to the special case k = 1.Comment: 17 pages, 1 figur
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
Counting Perron Numbers by Absolute Value
We count various classes of algebraic integers of fixed degree by their
largest absolute value. The classes of integers considered include all
algebraic integers, Perron numbers, totally real integers, and totally complex
integers. We give qualitative and quantitative results concerning the
distribution of Perron numbers, answering in part a question of W. Thurston.Comment: This work represents, in part, the PhD thesis of the second autho
Abelian Spiders
If G is a finite graph, then the largest eigenvalue L of the adjacency matrix
of G is a totally real algebraic integer (L is the Perron-Frobenius eigenvalue
of G). We say that G is abelian if the field generated by L^2 is abelian. Given
a fixed graph G and a fixed set of vertices of G, we define a spider graph to
be a graph obtained by attaching to each of the chosen vertices of G some
2-valent trees of finite length. The main result is that only finitely many of
the corresponding spider graphs are both abelian and not Dynkin diagrams, and
that all such spiders can be effectively enumerated; this generalizes a
previous result of Calegari, Morrison, and Snyder. The main theorem has
applications to the classification of finite index subfactors. We also prove
that the set of Salem numbers of "abelian type" is discrete.Comment: This work represents, in part, the PhD thesis of the second autho
Approximating the Largest Root and Applications to Interlacing Families
We study the problem of approximating the largest root of a real-rooted
polynomial of degree using its top coefficients and give nearly
matching upper and lower bounds. We present algorithms with running time
polynomial in that use the top coefficients to approximate the maximum
root within a factor of and when and respectively. We also prove corresponding
information-theoretic lower bounds of and
, and show strong lower
bounds for noisy version of the problem in which one is given access to
approximate coefficients.
This problem has applications in the context of the method of interlacing
families of polynomials, which was used for proving the existence of Ramanujan
graphs of all degrees, the solution of the Kadison-Singer problem, and bounding
the integrality gap of the asymmetric traveling salesman problem. All of these
involve computing the maximum root of certain real-rooted polynomials for which
the top few coefficients are accessible in subexponential time. Our results
yield an algorithm with the running time of for all
of them
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