On the complexity of invariant polynomials under the action of finite reflection groups

Abstract

Let be a finite reflection group and K[ 1 , . . . , ] be a multivariate polynomial ring over a field K. Let K[ 1 , . . . , ] be a set containing all invariant polynomials under the action of . Then the Chevalley-Shephard-Todd theorem states that there exists a sequence of homogeneous polynomials 1 , . . . , such that for any polynomial in K[ 1 , . . . , ] , there exists a unique polynomial new in K[ 1 , . . . , ], where 1 , . . . , are new variables, such that new ( 1 , . . . , ) = ( 1 , . . . , ). In this paper, we study the arithmetic complexity to compute new knowing and ( 1 , . . . , ). Our algorithm works for any finite reflection group and any set of generators ( 1 , . . . , ). Previously such a result was only known for a symmetric group or a directed product of symmetric groups and 1 , . . . , elementary symmetric functions.

Linear slices of hyperbolic polynomials and positivity of symmetric polynomial functions

“…, e n ) of f in terms of elementary symmetric polynomials. This can be, for example, obtained via Gröbner bases (see Proposition 1 in §1 of Chapter 7 in [7]) or alternatively with the algorithm presented in [27]. Secondly, once g ∈ R[e 1 , .…”