639 research outputs found
Localization of the complex zeros of parametrized families of polynomials
AbstractLet Pn(x)=xm+pm−1(n)xm−1+⋯+p1(n)x+pm(n) be a parametrized family of polynomials of a given degree with complex coefficients pk(n) depending on a parameter n∈Z≥0. We use Rouché’s theorem to obtain approximations to the complex roots of Pn(x). As an example, we obtain approximations to the complex roots of the quintic polynomials Pn(x)=x5+nx4−(2n+1)x3+(n+2)x2−2x+1 studied by A. M. Schöpp
Families of L-functions and their Symmetry
In [90] the first-named author gave a working definition of a family of
automorphic L-functions. Since then there have been a number of works [33],
[107], [67] [47], [66] and especially [98] by the second and third-named
authors which make it possible to give a conjectural answer for the symmetry
type of a family and in particular the universality class predicted in [64] for
the distribution of the zeros near s=1/2. In this note we carry this out after
introducing some basic invariants associated to a family
Stable Complete Intersections
A complete intersection of n polynomials in n indeterminates has only a
finite number of zeros. In this paper we address the following question: how do
the zeros change when the coefficients of the polynomials are perturbed? In the
first part we show how to construct semi-algebraic sets in the parameter space
over which all the complete intersection ideals share the same number of
isolated real zeros. In the second part we show how to modify the complete
intersection and get a new one which generates the same ideal but whose real
zeros are more stable with respect to perturbations of the coefficients.Comment: 1 figur
Random matrices, log-gases and Holder regularity
The Wigner-Dyson-Gaudin-Mehta conjecture asserts that the local eigenvalue
statistics of large real and complex Hermitian matrices with independent,
identically distributed entries are universal in a sense that they depend only
on the symmetry class of the matrix and otherwise are independent of the
details of the distribution. We present the recent solution to this
half-century old conjecture. We explain how stochastic tools, such as the Dyson
Brownian motion, and PDE ideas, such as De Giorgi-Nash-Moser regularity theory,
were combined in the solution.
We also show related results for log-gases that represent a universal model
for strongly correlated systems. Finally, in the spirit of Wigner's original
vision, we discuss the extensions of these universality results to more
realistic physical systems such as random band matrices.Comment: Proceedings of ICM 201
Unification of Residues and Grassmannian Dualities
The conjectured duality relating all-loop leading singularities of n-particle
N^(k-2)MHV scattering amplitudes in N=4 SYM to a simple contour integral over
the Grassmannian G(k,n) makes all the symmetries of the theory manifest. Every
residue is individually Yangian invariant, but does not have a local space-time
interpretation--only a special sum over residues gives physical amplitudes. In
this paper we show that the sum over residues giving tree amplitudes can be
unified into a single algebraic variety, which we explicitly construct for all
NMHV and N^2MHV amplitudes. Remarkably, this allows the contour integral to
have a "particle interpretation" in the Grassmannian, where higher-point
amplitudes can be constructed from lower-point ones by adding one particle at a
time, with soft limits manifest. We move on to show that the connected
prescription for tree amplitudes in Witten's twistor string theory also admits
a Grassmannian particle interpretation, where the integral over the
Grassmannian localizes over the Veronese map from G(2,n) to G(k,n). These
apparently very different theories are related by a natural deformation with a
parameter t that smoothly interpolates between them. For NMHV amplitudes, we
use a simple residue theorem to prove t-independence of the result, thus
establishing a novel kind of duality between these theories.Comment: 56 pages, 11 figures; v2: typos corrected, minor improvement
Analytic Solution of Bremsstrahlung TBA II: Turning on the Sphere Angle
We find an exact analytical solution of the Y-system describing a cusped
Wilson line in the planar limit of N=4 SYM. Our explicit solution describes
anomalous dimensions of this family of observables for any value of the `t
Hooft coupling and arbitrary R-charge L of the local operator inserted on the
cusp in a near-BPS limit. Our finding generalizes the previous results of one
of the authors & Sever and passes several nontrivial tests. First, for a
particular case L=0 we reproduce the predictions of localization techniques.
Second, we show that in the classical limit our result perfectly reproduces the
existing prediction from classical string theory. In addition, we made a
comparison with all existing weak coupling results and we found that our result
interpolates smoothly between these two very different regimes of AdS/CFT. As a
byproduct we found a generalization of the essential parts of the FiNLIE
construction for the gamma-deformed case and discuss our results in the
framework of the novel -formulation of the spectral problem.Comment: 39 pages, 4 figures; v2: minor corrections, references added; v3:
typos fixed, references updated; v4: typos fixe
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