Conference abstracts

Session C6 - Real-Number Complexity

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On the number of tangents to hypersurfaces in $\mathbb{R}P^n$ in random position

Khazhgali Kozhasov

SISSA, Italy   -   kkozhasov@sissa.it

Given $2n-2$ smooth hypersurfaces $X_1,\dots, X_{2n-2}$ in $\mathbb{R}P^n$ we are interested in the average number $T(X_1,\dots,X_{2n-2})$ of projective lines simultaneously tangent to $g_1X_1,\dots,g_{2n-2}X_{2n-2}$, where $g_1,\dots,g_{2n-2}$ are independent and uniformly distributed $O(n+1)$-transformations.

We express $T(X_1,\dots,X_{2n-2})$ in terms of the expected degree of the grassmanian $Gr(2,n+1)$ (introduced recently by P. Burgisser and A. Lerario) and some curvature integrals of $X_1,\dots,X_{2n-2}$.

For example, when $n=3$ and all $X_1,\dots,X_4$ are spheres (in the metric of $\mathbb{R}P^3$) of radius $r\in (0,\frac{\pi}{2})$ we have

$$ T(X_1,\dots,X_4) \approx 1.72 \left( \frac{8}{\pi} \sin r \cos r \right)^4$$

Joint work with Antonio Lerario (SISSA).

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FoCM 2017, based on a nodethirtythree design.