A natural basis for intersection numbers

Understand

We advertise elementary symmetric polynomials $e_i$ as the natural basis for generating series $A_{g,n}$ of intersection numbers of genus g and n marked points.

• Closed formulae for $A_{g,n}$ are known for genera $0$ and $1$ -- this approach provides formulae for $g = 2,3,4$, together with an algorithm to compute the formula for any g.
• The claimed naturality of the e_i basis relies in the unexpected vanishing of some coefficients with a clear pattern: we conjecture that $A_{g,n}$ can have at most $g$ factors $e_i$, with $i>1$, in its expansion.
• This observation promotes a paradigm for more general cohomology classes.