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

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 equivalent graphical characterization on the monomials in this Chow ring, as well as the characterization on the algebraic reduction on such monomials. Moreover, we provide an algorithm for computing the intersection degree of tuples of Keel's divisor classes — we call it the forest algorithm; the complexity of which is $O\left(n^3\right)$ in the worst case, where n refers to the number of marks in the ambient moduli space. Last but not least, we introduce three identities on multinomial coefficients which naturally came into play, showing that they are all equivalent to the correctness of the base case of the forest algorithm.