Enumerating Permutation Polynomials over finite fields by degree

We prove an asymptotic formula for the number of permutation for which the associated permutation polynomial has degree smaller than $q-2$.Comment: LaTeX2e amsart 5 page

Pseudo-elliptic integrals, units, and torsion

We remark on pseudo-elliptic integrals and on exceptional function fields, namely function fields defined over an infinite base field but nonetheless containing non-trivial units. Our emphasis is on some elementary criteria that must be satisfied by a squarefree polynomial whose square root generates a quadratic function field with non-trivial unit. We detail the genus 1 case.Comment: Submitted preprin

Divisibility of reduction in groups of rational numbers

Given a multiplicative group of non zero rational numbers and a positive integer m , we consider the problem of determining the density of the set of primes p for which the order of the reduction modulo p of the group is divisible by m . In the case when the group is nitely generated the density is explicitly computed. Some example of groups with innite rank are considered

Entanglement dynamics and chaos in long-range quantum systems

Over the past twenty years, experimental and technological progresses have motivated a renewed attention to the study of non-equilibrium isolated many-body systems, leading to a relatively well-established paradigm in the case of local Hamiltonians. In the present thesis, I have used quantum information theoretical tools to study out-of-equilibrium dynamics, with particular attention on long-range interacting many-body systems. I have explored the dynamics of bipartite and multipartite entanglement in connection to chaos and scrambling in various long-range (clean and disordered) models. The results contained in this thesis contribute to establishing semi-classical tools as powerful techniques for the description of the quantum information spreading in long-range systems. I have further considered a different, yet connected question, concerning the multipartite entanglement structure of chaotic eigenstates and its generic evolution

Quantum bounds on the generalized Lyapunov exponents

We discuss the generalized quantum Lyapunov exponents $L_q$, defined from the growth rate of the powers of the square commutator. They may be related to an appropriately defined thermodynamic limit of the spectrum of the commutator, which plays the role of a large deviation function, obtained from the exponents $L_q$ via a Legendre transform. We show that such exponents obey a generalized bound to chaos due to the fluctuation-dissipation theorem, as already discussed in the literature. The bounds for larger $q$ are actually stronger, placing a limit on the large deviations of chaotic properties. Our findings at infinite temperature are exemplified by a numerical study of the kicked top, a paradigmatic model of quantum chaos.Comment: The paper is dedicated to Professor Giulio Casati on the Occasion of his 80th Birthday, submitted to the Special Issue of Entropy: "Quantum chaos - dedicated to Professor Giulio Casati on the Occasion of His 80th Birthday

On ther-rank Artin Conjecture, II

AbstractFor any finitely generated subgroupΓofQ* we compute a formula for the density of the primes for which the reduction modulopofΓcontains a primitive root modulop. We use this to conjecture a characterization of “optimal” subgroups (i.e., subgroups that have maximal density). We also improve the error term in the asymptotic formula of Pappalardi's Theorem 1.1 (Math. Comp.66(1997), 853–868)