The aim of the talk is to explain how to make these statements precise and prove them, using differential Galois theory. The reference to all of the material here is [vdPS03]. By solving a differential equation we mean obtaining a solution via a finite number of operations of the following kind (starting with a rational function): • Adding a function algebraic over the functions we already have...

Starting from a very general trace-form entropy, we introduce a pair of algebraic structures endowed by a generalized sum and a generalized product. These algebras form, respectively, two Abelian fields in the realm of the complex numbers isomorphic each other. We specify our results to several entropic forms related to distributions recurrently observed in social, economical, biological and ph...

Let π be a regular algebraic cuspidal automorphic representation of GL2 over an imaginary quadratic number field K, and let ` be a prime number. Assuming the central character of π is invariant under the non-trivial automorphism of K, it is shown that there is a continuous irreducible `-adic representation ρ of Gal(K/K) such that L(s, ρv) = L(s, πv) whenever v is a prime of K outside an explici...

where f is a polynomial of degree e in n variables with coefficients in a finite field F of characteristic p, E is an overfield of F, and ψ is an additive character of F. It is interesting to ask what happens with sums S(E, f) when one varies E, or when one varies f . In order to talk precisely about the latter, we denote by P(e, n) the scheme parameterizing all polynomials in n variables of de...

Let O be an order of an algebraic number field. It was shown by Ge that given a factorization of an O-ideal a into a product of O-ideals it is possible to compute in polynomial time an overorder O′ of O and a gcdfree refinement of the input factorization; i.e., a factorization of aO′ into a power product of O′-ideals such that the bases of that power product are all invertible and pairwise copr...

We investigate valued fields which admit a valuation basis. Given a countable ordered abelian group G and a real closed, or algebraically closed field F , we give a sufficient condition for a valued subfield of the field of generalized power series F ((G)) to admit a K-valuation basis. We show that the field of rational functions F (G) and the field F (G) of power series in F ((G)) algebraic ov...

In this paper we investigate the simulation of real and reactive power spot markets. While spot pricing of real power remains a viable option for the creation of a power system market, the future of a reactive power spot market remains cloudy. The large capital investment portion needed in pricing reactive power as well as the highly volatile nature of reactive power spot prices makes the creat...

Let π be a regular algebraic cuspidal automorphic representation of GL2 over an imaginary quadratic number field K, and let l be a prime number. Assuming the central character of π is invariant under the non-trivial automorphism of K, it is shown that there is a continuous irreducible l-adic representation ρ of Gal(K/K) such that L(s, ρv) = L(s, πv) whenever v is a prime of K outside an explici...

We investigate valued fields which admit a valuation basis. Given a countable ordered abelian group G and a real closed or algebraically closed field F with subfield K, we give a sufficient condition for a valued subfield of the field of generalized power series F ((G)) to admit aK-valuation basis. We show that the field of rational functions F (G) and the field F (G) of power series in F ((G))...