Bounds for gradient trajectories of polynomial and definable functions with applications

We show that gradient trajectories of a denable (in an o-minimal structure) family of functions are of uniformly bounded length. We show 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) = (n)((3d 4)n1 + 2(3d 3)n2) and (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 curves (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 quantitative Morse-Sard Theorem