Séminaire Lotharingien de Combinatoire, 82B.72 (2019), 12 pp.

David Harry Richman

The distribution of Weierstrass points on a tropical curve

Abstract. The set of (higher) Weierstrass points on a metric graph of genus g > 1 is an analogue of the set of N-torsion points on a circle. As N grows, the torsion points "distribute evenly" over a circle. This makes it natural to ask how Weierstrass points distribute on a graph, as the degree of the corresponding divisor grows. We study how Weierstrass points behave on tropical curves (i.e. finite metric graphs) in analogy with complex algebraic curves (i.e. Riemann surfaces), and explain how their distribution can be described in terms of electrical networks. This is a tropical analogue of a result of Neeman, for the distribution of Weierstrass points on a compact Riemann surface, and extends previous work of Zhang and Amini on the non-Archimedean case.

Received: November 15, 2018. Accepted: February 17, 2019. Final version: April 1, 2019.

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