On the automorphisms of generalized algebraic geometry codes

Designs, Codes and Cryptography - We consider the class of generalized algebraic geometry codes (GAG codes) formed by two collections of places, with places of the same degree in each collection....     

On algebraic automorphisms and their rational invariants

Let X be an affine irreducible variety over an algebraically closed field k of characteristic zero. Given an automorphism Φ, we denote by k(X)^Φ its field of invariants, i.e., the set of rational functions f on X such that f o Φ = f. Let n(Φ) be the transcendence degree of k(X)^Φ over k. In this paper we study the class of automorphisms Φ of X for which n(Φ) = dim X - 1. More precisely, we show that under some conditions on X, every such automorphism is of the form Φ = ϕ_g, where ϕ is an algebraic action of a linear algebraic group G of dimension 1 on X, and where g belongs to G. As an application, we determine the conjugacy classes of automorphisms of the plane for which n(Φ) = 1.     

On the Galois group of some Fuchsian systems

In this paper, we give a new result ofn the differential Galois theory of linear ordinary differential equations. In particular, we compute the differential Galois group for a special type of nonresonant Fuchsian system.     

On the group of conjugating automorphisms of a free group

The group of conjugating automorphisms of a free group and certain subgroups of this group, namely, the group of McCool basis-conjugating automorphisms and the Artin braid group are considered. The Birman theorem on the representation of a braid group by matrices is sharpened. Translated fromMatematicheskie Zametki, Vol. 60, No. 1, pp. 92–108, July, 1996.     

Exceptional group ring automorphisms for some metabelian groups

Let H be a generalized dihedral, semi-dihedral, quaternion, or modular group, and let A = (u, v, w) be a product of three odd order cyclic groups, with (|v|,|w|) = 1. For R a semi-local Dedekind domain of characteristic 0 in which no prime divisor of |H|.|A| is invertible, we prove that there is a semi-direct product G = H × A such that the group ring RG has an exceptional automorphism, i.e. provides a counter-example to a well-known conjecture of Zassenhaus on automorphisms of group rings     

On the automorphisms of the spectral completion of the algebraic numbers field

Let be the absolute Galois group of Q and let A=C(G,C) be the Banach algebra of all continuous functions defined on G with values in C. Let be the conjugation automorphism of C and let B be the R-Banach subalgebra of A consisting of continuous functions f such that for all σ∈G. Let ‖x‖=sup{|σ(x)|:σ∈G} be the spectral norm on and let be the spectral completion of . Using a canonical isometry between and B we study the structure of the group of R-algebras automorphisms of and the structure of its subgroup of all automorphisms of which when restricted to give rise to elements of G. We introduce a topology on and prove that this last one is homeomorphic and group isomorphic to G.     

On the Brauer group of a semisimple algebraic group

Let k be a field with algebraic closure , G a semisimple algebraic k-group, and a maximal torus with character group X(T). Denote Λ the abstract weight lattice of the roots system of G, and by and the n-torsion subgroup of the Brauer group of k and G, respectively. We prove that if chark does not divide n and n is prime to the order of Λ/X(T) then the natural homomorphism is an isomorphism.     

On the group of holomorphic automorphisms of ℂ n

Oblatum 21-IX-1991 & 30-IV-1992     

Linear systems: some algebraic aspects

In the past twenty years mathematically minded engineers working in control theory have discovered a number of results of a purely algebraic nature. The present article aims at giving a survey of this area for readers interested in linear algebra but without previous knowledge of this topic.     

Birational automorphisms of three-dimensional algebraic varieties

The survey is devoted to the birational theory of three-dimensional algebraic Fano varieties.Translated from Itogi Nauki i Tekhniki, Sovremennye Problemy Matematiki, Vol. 12, pp. 159–236, 1979.     

Plane algebraic curves with Singer automorphisms

Non-singular plane algebraic curves over with a Singer group of PGL(3,q) in their automorphism group are classified. Apart from three distinguished points, the set of -rational points of such curves can be partitioned into 2−(q²+q+1,q+1,1) designs each isomorphic to the finite projective plane .     

The algebraic entropy of the special linear character automorphisms of a free group on two generators

In this note, we establish a connection between the dynamical degree, or algebraic entropy of a certain class of polynomial automorphisms of , and the maximum topological entropy of the action when restricted to compact invariant subvarieties. Indeed, when there is no cancellation of leading terms in the successive iterates of the polynomial automorphism, the two quantities are equal. In general, however, the algebraic entropy overestimates the topological entropy. These polynomial automorphisms arise as extensions of mapping class actions of a punctured torus on the relative -character varieties of embedded in . It is known that the topological entropy of these mapping class actions is maximized on the relative character variety comprised of reducible characters (those whose boundary holonomy is ). Here we calculate the algebraic entropy of the induced polynomial automorphisms on the character varieties and show that it too solely depends on the topology of .

     

On automorphisms for some fibered incidence groups

Summary In any kinematic space (with non trivial fibration) the group of all automorphisms of the geometric structure which preserve both parallelisms is shown to consist exactly of those automorphisms of the algebraic structure which preserve the fibration. Moreover a characterization of such group is given for a particular class of kinematic spaces. Research supported by the Italian Ministry of Education (40% M.P.I.).