# Positivity in classical enumerative geometry: a case study in synchronized AI-assisted mathematics

arXiv: https://arxiv.org/abs/2605.25271 · HTML: https://mlsanity.com/paper/2605.25271/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 1 of its 24 references that are in the map.
- First posted on arXiv 24 May 2026.

## Builds on (most cited first, 1 of 1)

- [AlphaEvolve: A coding agent for scientific and algorithmic discovery](https://mlsanity.com/paper/2506.13131/) (2025)

## Abstract (arXiv)

We study the symmetric polynomial prod_α∈ An,dbigl(1+α1 x1+cdots+αn xnbigr) where An,d:=α∈mathbbZ≥ 0n:|α|=d, which is the total Chern class of Symd(mathbbCn), viewed as a torus representation whose Chern roots are the weights α1 x1+cdots+αn xn for α∈ An,d. Its homogeneous degree-k part ck(n,d) is the k-th Chern class of Symd(mathbbCn). These Chern classes, together with their coefficients in various symmetric function bases, play a central role in enumerative geometry. Despite their simple definition, general closed formulas for their coefficients are subtle, and many structural properties of these classes have remained poorly understood. In this paper we prove several conjectures concerning their structure, establish explicit formulas, and study log-concavity properties for both the Chern classes and their K-theoretic analogue. In rank two, passing to the Schur basis and expanding the Schur coefficients in the binomial basis of d, we uncover a new binomial log-concavity phenomenon and prove refined positivity results. The paper demonstrates a novel methodology: we combine several AI systems with human mathematical insight in a coordinated workflow, deploying each tool according to its strengths in experimental discovery, conjecture formation, symbolic proof construction, and verification. To our knowledge, this is one of the first detailed case studies of orchestrating multiple AI tools to make substantial progress on a coherent mathematical research project.