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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.