Armin Rainer

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We study the regularity of the roots of complex monic polynomials P (t) of degree n depending smoothly on a real parameter t. If P (t) is C∞ and no two of the continuously chosen roots meet of infinite order of flatness, then there exists a locally absolutely continuous parameterization of the roots. Simple examples show that the conclusion is best(More)
We prove that the roots of a definable C∞ curve of monic hyperbolic polynomials admit a definable C∞ parameterization, where ‘definable’ refers to any fixed o-minimal structure on (R,+, ·). Moreover, we provide sufficient conditions, in terms of the differentiability of the coefficients and the order of contact of the roots, for the existence of Cp (for p ∈(More)
Let Pa(Z) = Z n + ∑n j=1 ajZ n−j be a C curve of monic polynomials, ai ∈ C(I,C) where I ⊂ R is an interval. We show that if k = k(n) is sufficiently large then any choice of continuous roots of Pa is locally absolutely continuous, in a uniform way with respect to maxj ‖aj‖Ck on compact subintervals. This solves a problem that was open for more then a decade(More)
P (t)(x) = x n − a 1 (t)x n−1 + · · · + (−1) n a n (t) mit ausschließlich reellen Wurzeln (später werden wir solche Polynome 'hyper-bolisch' nennen), welche durch t nahe 0 in R glatt parametrisiert ist. Können wir n glatte Funktionen x 1 (t),. .. , x n (t) finden, die die Wurzeln von P (t) für jedes t parametrisieren? (2002).
The orbit projection π :M →M/G of a proper G-manifold M is a fibration if and only if all points inM are regular. Under additional assumptions we show that π is a quasifibration if and only if all points are regular. We get a full answer in the equivariant category: π is a G-quasifibration if and only if all points are regular.
We prove the exponential law A(E × F,G) ∼= A(E,A(F,G)) (bornological isomorphism) for the following classes A of test functions: B (globally bounded derivatives), W∞,p (globally p-integrable derivatives), S (Schwartz space), B[M ] (globally Denjoy–Carleman), W [M ],p (Sobolev– Denjoy–Carleman), and S [M ] [L] (Gelfand–Shilov). Here E,F,G are convenient(More)
Let P (x)(z) = zn + Pn j=1(−1) aj(x)z be a family of polynomials of fixed degree n whose coefficients aj are germs at 0 of smooth (C ) complex valued functions defined near 0 ∈ Rq. We show: If P is generic there exists a finite collection T of transformations Ψ : Rq, 0 → Rq, 0 such that S {im(Ψ) : Ψ ∈ T } is a neighborhood of 0 and, for each Ψ ∈ T , the(More)