A+ CATEGORY SCIENTIFIC UNIT

PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

Bounding the length of gradient trajectories

Volume 127 / 2021

Didier D’Acunto, Krzysztof Kurdyka Annales Polonici Mathematici 127 (2021), 13-50 MSC: Primary 32Bxx, 34Cxx; Secondary 32Sxx, 14P10. DOI: 10.4064/ap200527-3-6 Published online: 9 August 2021

Abstract

We propose a method to bound the length of gradient trajectories by comparison with the length of corresponding talwegs (ridge or valley lines), and we obtain several applications. We show that gradient trajectories of a definable (in an o-minimal structure) family of functions are of uniformly bounded length. We prove that the length of a trajectory of the gradient of a polynomial in $n$ variables of degree $d$ in a ball of radius $r$ is bounded by $rA(n,d)$, where $A(n,d)=\nu (n)((3d-4)^{n-1} + 2(3d-3)^{n-2})$ and $\nu (n)$ is an explicit constant. We give explicit bounds for the length of gradient trajectories of quasipolynomials and trigonometric quasipolynomials. As an application we give a construction of a curve (piecewise gradient trajectory of a polynomial) joining two points in an open connected semialgebraic set. We give an explicit bound for its length. We also obtain an explicit and quite sharp bound in Yomdin’s version of the quantitative Morse–Sard theorem.

Authors

  • Didier D’AcuntoLaboratoire de Mathématiques, UMR 5127 CNRS
    Université Savoie Mont Blanc
    Campus Scientifique
    F-73376 Le Bourget du Lac Cedex, France
    e-mail
  • Krzysztof KurdykaLaboratoire de Mathématiques, UMR 5127 CNRS
    Université Savoie Mont Blanc
    Campus Scientifique
    F-73376 Le Bourget du Lac Cedex, France
    e-mail

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image