Special Year Learning Seminar
On Common Roots of Legendre Polynomials
In 2011 Bourgain and Rudnick showed that if $\gamma$ is a curve of non-vanishing curvature on the 2d standard flat torus, then there are no Laplace eigenfunctions of arbitrarily large eigenvalues containing $\gamma$ in their nodal set. We show that if $\gamma$ is a small circle on the 2-sphere, then there are no zonal spherical harmonics of arbitrarily large eigenvalues containing $\gamma$ in their nodal set. Equivalently, there are only finitely many Legendre polynomials that vanish at any given non-zero root. Stieltjes conjecture (1890) says that only one Legendre polynomial can vanish at any given non-zero root. Time permits, we will also discuss quantitative bounds.
The talk is based on joint works with Borys Kadets and Adi Weller Weiser.