A tree-based algorithm for the integration of monomials in the Chow ring of the moduli space of stable marked curves of genus zero

TL;DR

This paper presents a tree-based algorithm for computing intersection degrees of divisor classes in the Chow ring of the moduli space of genus zero stable marked curves, along with complexity analysis and new identities on multinomial coefficients.

Contribution

It introduces a novel tree-based algorithm for intersection computations in the Chow ring and provides proofs for new multinomial coefficient identities.

Findings

Algorithm efficiently computes intersection degrees.

Complexity analysis shows the algorithm's practicality.

New multinomial identities are proven.

Abstract

The Chow ring of the moduli space of marked rational curves is generated by Keel's divisor classes. The top graded part of this Chow ring is isomorphic to the integers, generated by the class of a single point. In this paper, we give an algorithm for computing the intersection degree of tuples of Keel's divisor classes. This computation is a concrete but complicated algorithmic question in the field. Also, we give a simple complexity argument for the algorithm. Additionally, we introduce three identities on multinomial coefficients, as well as proofs for them.