An equivariant isomorphism theorem for mod p\mathfrak p reductions of arboreal Galois representations

Let $\phi$ be a quadratic, monic polynomial with coefficients in $\mathcal O_{F,D}[t]$, where $\mathcal O_{F,D}$ is a localization of a number ring $\mathcal O_F$. In this paper, we first prove that if $\phi$ is non-square and non-isotrivial, then there exists an absolute, effective constant $N_\phi$ with the following property: for all primes $\mathfrak p\subseteq\mathcal O_{F,D}$ such that the reduced polynomial $\phi_\mathfrak p\in (\mathcal O_{F,D}/\mathfrak p)[t][x]$ is non-square and non-isotrivial, the squarefree Zsigmondy set of $\phi_{\mathfrak p}$ is bounded by $N_\phi$. Using this result, we prove that if $\phi$ is non-isotrivial and geometrically stable then outside a finite, effective set of primes of $\mathcal O_{F,D}$ the geometric part of the arboreal representation of $\phi_{\mathfrak p}$ is isomorphic to that of $\phi$. As an application of our results we prove R. Jones' conjecture on the arboreal Galois representation attached to the polynomial $x^2+t$.Comment: Comments are welcome

Learning Curve and Wind Power

This study explores the reasons why countries have chosen subsidies to green electricity instead of implementing the more common Pigouvian tax on polluting emissions. I focus on the learning by doing effects from the production of wind power on the cost of future production as a justification for the observed policies. In doing so, I present two models that differ in the way I introduce learning. Under reasonable parameter values, the price paid to a firm for the energy produced from wind power is heterogeneous, and varies among the firms that produce energy from wind power according to the index of productivity of the firm itself. The suggested strategies of this research differ from the main price-driven schemes adopted by EU members; by comparing such results with European Union policy, the paper show that EU policy is not optimal.learning by doing, environmental policy, Pigouvian taxes, subsidies.

Eisenstein Polynomials over Function Fields

In this paper we compute the density of monic and non-monic Eisenstein polynomials of fixed degree having entries in an integrally closed subring of a function field over a finite field