A Difference Version of Nori's Theorem

We consider (Frobenius) difference equations over (F_q(s,t), phi) where phi fixes t and acts on F_q(s) as the Frobenius endomorphism. We prove that every semisimple, simply-connected linear algebraic group G defined over F_q can be realized as a difference Galois group over F_{q^i}(s,t) for some i in N. The proof uses upper and lower bounds on the Galois group scheme of a Frobenius difference equation that are developed in this paper. The result can be seen as a difference analogue of Nori's Theorem which states that G(F_q) occurs as (finite) Galois group over F_q(s).Comment: 29 page

The Fuchsian differential equation of the square lattice Ising model χ(3)\chi(3) susceptibility

Using an expansion method in the variables $x_i$ that appear in the $(n-1)$-dimensional integrals representing the $n$-particle contribution to the Ising square lattice model susceptibility $\chi$, we generate a long series of coefficients for the 3-particle contribution $\chi^{(3)}$, using a $N^4$ polynomial time algorithm. We give the Fuchsian differential equation of order seven for $\chi^{(3)}$ that reproduces all the terms of our long series. An analysis of the properties of this Fuchsian differential equation is performed.Comment: 15 pages, no figures, submitted to J. Phys.

Connected Linear Groups as Differential Galois Groups

this paper we give a proof of the followin

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In this paper we give a proof of the following THEOREM 1.1. Let C be an algebraically closed field of characteristic zero, G a connected linear algebraic group defined o�er C, and k a differential field containing C as its field of constants and of finite, nonzero transcendenc

Solvable-by-finite groups as differential Galois groups

We prove the inverse problem of differential Galois theory over the differential field k=C(x), where C is an algebraic closed field of characteristic zero, for linear algebraic groups G over CC with a solvable identity component G°. We show that for any k-irreducible principal homogeneous space V for G, the derivation d/dx of k can be extended on k(V) in such a way that k(V) is a Picard-Vessiot extension of k with Galois group G. The proof is constructive up to the finite embedding problem of classicalGalois theory over C(x)