Equidistribution in arithmetic geometry and dynamics Juan Rivera-Letelier U. of Rochester Parameter Problems in Analytic Dynamics Imperial College, June The probabilistic viewpoint in arithmetic geometry Roots of Littlewood and Einsenstein polynomials; Discrepancy; Equidistribution. Roots of Littlewood polynomials Figure : Roots of polynomials with coefficients +and -, by Tiozzo. Christensen, Derbyshire, Baez, ... Roots of Einsenstein polynomials Figure : Roots of monic polynomials of degree with coefficients or .
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Equidistribution in arithmetic geometryand dynamics
Juan Rivera-Letelier
U. of Rochester
Parameter Problems in Analytic DynamicsImperial College, June
The probabilistic viewpointin arithmetic geometry
Roots of Littlewood and Einsenstein polynomials;
Discrepancy;
Equidistribution.
Roots of Littlewood polynomials
Figure : Roots of polynomials with coefficients + and −, by Tiozzo.
Christensen, Derbyshire, Baez, ...
Roots of Einsenstein polynomials
Figure : Roots of monic polynomials of degree with coefficients or .
Roots of single Einsenstein polynomial Discrepancy
Theorem (Radial discrepancy)
P(z) = adzd +ad−zd−+ · · ·+a ∈C[z], ada , .
For every ε in (,), we have
d#
{α root of P : |α| < − ε or |α| >
− ε
}≤ ε
d log
∑dj= |aj |√|aad |
.Hughes–Nikeghbali, ;
Applications: Roots of Littlewood and Einsenstein polynomials(ε ∼ √