BOUNDS FOR GRADIENT TRAJECTORIES AND GEODESIC DIAMETER OF REAL ALGEBRAIC SETS
D. D'ACUNTO and K. KURDYKA


Let $M\subset \mathbb{R}^n$ be a connected component of an algebraic set $\varphi^{-1}(0)$, where $\varphi$ is a polynomial of degree $d$. Assume that $M$ is contained in a ball of radius $r$. We prove that the geodesic diameter of $M$ is bounded by $2r\nu(n)d(4d-5)^{n-2}$, where $\nu(n)=2{\Gamma({1}/{2})\Gamma(({n+1})/{2})}{\Gamma({n}/{2})}^{-1}$. This estimate is based on the bound $r\nu(n)d(4d-5)^{n-2}$ for the length of the gradient trajectories of a linear projection restricted to $M$.

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туда же много интересного про "управляемую трансверсальность" и оценку диаметра в терминах степени тут

https://projecteuclid.org/journals/journal-of-differential-geometry/volume-44/issue-4/Symplectic-submanifolds-and-almost-complex-geometry/10.4310/jdg/1214459407.full

Symplectic submanifolds and almost-complex geometry
S. K. Donaldson