A generalization of Puiseux’s theorem and lifting curves over invariants
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- Losik, M., Michor, P.W. & Rainer, A. Rev Mat Complut (2012) 25: 139. doi:10.1007/s13163-011-0062-y
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Let ρ:G→GL (V) be a rational representation of a reductive linear algebraic group G defined over ℂ on a finite dimensional complex vector space V. We show that, for any generic smooth (resp. CM) curve c:ℝ→V//G in the categorical quotient V//G (viewed as affine variety in some ℂn) and for any t0∈ℝ, there exists a positive integer N such that t↦c(t0±(t−t0)N) allows a smooth (resp. CM) lift to the representation space near t0. (CM denotes the Denjoy–Carleman class associated with M=(Mk), which is always assumed to be logarithmically convex and derivation closed). As an application we prove that any generic smooth curve in V//G admits locally absolutely continuous (not better!) lifts. Assume that G is finite. We characterize curves admitting differentiable lifts. We show that any germ of a C∞ curve which represents a lift of a germ of a quasianalytic CM curve in V//G is actually CM. There are applications to polar representations.