The transcendental Bézout problem revisited

Abstract: Bézout's classical theorem states that n complex polynomials of degree k on C^n have at most k^n isolated common zeros. The logarithm of the maximal function of an entire function on C, instead of the degree, controls the number of zeros in a ball of radius r. The transcendental Bézout problem seeks to extend this estimate to entire self-mappings f of C^n via the n-th power of the logarithm of the maximal function. A celebrated counterexample of Cornalba-Shiffman shows that this is dramatically false for n>1. However, it is true on average, under lower bounds on the Jacobian, or in a weaker form for small constant perturbations of f. We explain how topological considerations of persistent homology and Morse theory shed new light on this question proving the expected bound for a robust count of zeros. This is part of a larger joint project with Buhovsky, Payette, Polterovich, Polterovich, and Stojisavljevic.

differential geometrydynamical systemsgeometric topologysymplectic geometry

Audience: researchers in the topic

Geometry and Dynamics seminar

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