arXiv subjects
Publications and source records attributed to Gregor Podlogar.
We study so called weakly-periodic twisted-multiplicative automorphisms of the free skew-field. In particular, we show that any automorphism of a free skew-field that is defined by a periodic automorphism of a free group is equivalent to a diagonal automorphism.
We consider the Noether's problem on the noncommutative real rational functions invariant under the linear action of a finite group. For abelian groups the invariant skew-fields are always rational. We show that for a solvable group the invariant skew-field is finitely generated. The skew-field invariant under a linear action of a solvable group is rational if the action is well-behaved -- given by a so-called complete representation. We determine the groups that admit such representations and call them totally psuedo-unramified. In the second part we study the reach of totally psuedo-unramified groups and classify totally pseudo-unramified p-groups of rank at most 5.
This paper describes the structure of invariant skew fields for linear actions of finite solvable groups on free skew fields in $d$ generators. These invariant skew fields are always finitely generated, which contrasts with the free algebra case. For abelian groups or solvable groups $G$ with a well-behaved representation theory it is shown that the invariant skew fields are free on $|G|(d-1)+1$ generators. Finally, positivity certificates for invariant rational functions in terms of sums of squares of invariants are presented.