Noncommutative rational functions invariant under the action of a finite solvable group

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.