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

Technical Summary: Positivity in Classical Enumerative Geometry

Problem Statement
The paper investigates the structural properties of the total Chern class of the symmetric power of a vector bundle, specifically $Sym_d(\mathbb{C}^n)$. Viewed as a torus representation, its total Chern class is given by the product:
$$c(Sym_d E) = \prod_{\alpha \in A_{n,d}} (1 + \alpha_1 x_1 + \cdots + \alpha_n x_n),$$
where $A_{n,d} = \{\alpha \in \mathbb{Z}_{\ge 0}^n : |\alpha| = d\}$. The homogeneous degree-$k$ part, denoted $c_k(n, d)$, is a symmetric polynomial. While these classes are central to enumerative geometry (e.g., calculating degrees of Fano schemes), general closed formulas for their coefficients in standard symmetric function bases (elementary, Schur) are subtle and often lack clear structural patterns.

The authors focus on two primary questions:

1. Polynomiality: Do the coefficients of $c_k(n, d)$ in the elementary symmetric basis exhibit a specific polynomial dependence on the parameters $n$ (rank) and $d$ (degree)?
2. Positivity and Log-Concavity: Do these coefficients, or their expansions in other bases, satisfy positivity and log-concavity properties? Specifically, the paper examines whether Schur coefficients, when expanded in a binomial basis with respect to $d$, yield non-negative and log-concave sequences.

Methodology: Synchronized AI-Assisted Mathematics
A defining feature of this work is its methodology, described as a "synchronized AI-assisted" workflow. The authors did not treat AI as a monolithic black box but orchestrated multiple systems with distinct roles:

• AlphaEvolve (Google DeepMind): Used as an evolutionary coding agent to discover closed formulas for low-degree cases (specifically $c_2(n, d)$) and to act as a "detective" for structural recurrences in coefficient tables. It identified approximate identities and candidate recurrence relations that guided human intuition.
• ChatGPT 5.5 Pro: Served as an interactive assistant for symbolic derivations. It successfully derived the strong binomiality theorem, provided proofs for the $j=0$ and $j=1$ cases of binomial positivity, and generated the K-theoretic analogue.
• Co-Mathematician (Google DeepMind): Deployed in the final stage to handle the most difficult proofs (specifically $j=2$ for positivity and $j=1$ for log-concavity) which previous tools could not resolve. It produced proof drafts based on the structural hints found by AlphaEvolve.
• Human Insight: Mathematicians selected the problems, formulated the conjectures based on AI-generated data, changed the basis (from Schur to binomial) to reveal hidden patterns, and verified the final rigorous proofs.

Key Contributions and Results

1. Strong Binomiality Theorem (Section 2)
The paper proves a "strong binomiality" conjecture regarding the coefficients of $c_k(n, d)$ in the elementary symmetric basis.

• Result: For any fixed $k$, the coefficients $f_\lambda(n, d)$ in the expansion $c_k(n, d) = \sum f_\lambda(n, d) e_\lambda$ lie in the polynomial ring generated by a specific set of binomial coefficients:
  $$T_r(n, d) = \binom{d+n-1}{n+r-1}, \quad 1 \le r \le k.$$
• Significance: This establishes that the coefficients are not merely polynomial in $n$ and $d$ separately, but are polynomials in a distinguished family of binomial coefficients. The proof utilizes factorial moments of weak compositions and Newton identities.
• K-Theory Analogue: The authors establish a multiplicative K-theoretic analogue, $NN_{n,d}$, showing that its diagonal coefficients recover the Chern classes and that its off-diagonal coefficients share the same binomial dependence structure.

2. Rank-Two Binomial Positivity and Log-Concavity (Section 3 & 5)
The paper focuses on the rank-two case ($n=2$), expanding the Chern classes in the Schur basis $s_{(k-j, j)}$ and then expanding the Schur coefficients $A_{k,j}(d)$ in the binomial basis of $d$:
$$A_{k,j}(d) = \sum_r B_{k,j,r} \binom{d}{r}.$$
The authors formulate and prove two conjectures for the first few modes ($j$):

• Conjecture 3.2 (Binomial Positivity): $B_{k,j,r} \ge 0$.
• Conjecture 3.3 (Binomial Log-Concavity): The sequence $(B_{k,j,r})_r$ is log-concave ($B_{k,j,r}^2 \ge B_{k,j,r-1}B_{k,j,r+1}$).

Proofs:

• Theorem 3.4: Establishes that binomial log-concavity implies ordinary log-concavity of the value sequence $A_{k,j}(d)$.
• Theorem 3.6 & 3.7: Prove positivity and log-concavity for $j=0$ and $j=1$ using combinatorial interpretations involving fixed-point-free permutations with prescribed cycle defects.
• Theorem 5.2: Proves binomial positivity for $j=2$. The proof relies on a recurrence relation derived from the structure of the Chern classes, expressing $\Delta A_{k,2}$ in terms of lower modes ($j=0, 1$) and linear factors, leveraging the already established positivity of lower modes.
• Theorem 5.4: Proves binomial log-concavity for $j=1$. This involves expressing the coefficients in terms of permutation numbers $D(N, K)$ and reducing the inequality to a finite check of polynomial non-negativity via Abel summation.

3. Plücker Coefficients and Shifted Log-Concavity (Section 6)
The authors extend the inquiry to Plücker coefficients of coincident root strata, which count lines with prescribed intersection multiplicities on hypersurfaces.

• Observation: Unlike the Chern class case, the ordinary binomial expansion of these polynomials can have negative coefficients. However, after shifting the degree variable $d$ by the codimension $N = |\lambda|$, the coefficients in the shifted binomial basis appear positive and log-concave.
• Conjecture 6.1: Proposes that the shifted-binomial coefficients for Plücker polynomials are non-negative and log-concave.
• Status: The paper provides a proof strategy based on recursive formulas for equivariant cohomology classes but identifies a "missing step": a positive transition formula (kernel) that preserves log-concavity under the summation over Schur indices. The authors note that standard real-rootedness arguments fail for these cases.

Significance and Claims
The paper claims to be one of the first detailed case studies of orchestrating multiple AI tools to make substantial progress on a coherent mathematical research project.

• Mathematical Contribution: It resolves long-standing conjectures regarding the polynomial structure of Chern class coefficients and establishes new positivity and log-concavity phenomena in rank two. It provides explicit formulas and recursive structures previously unknown.
• Methodological Contribution: It demonstrates a workflow where AI tools are assigned specific roles (formula discovery, recurrence detection, proof drafting) and human mathematicians provide the strategic direction, basis changes, and final verification. The authors emphasize that the human role is critical in "changing perspective" (e.g., shifting to the binomial basis) to reveal patterns that AI alone might not have identified.
• Modesty: The authors explicitly state that higher-mode versions of the rank-two log-concavity theorem remain open, and the Plücker extension is presented as a conjectural direction with a known obstruction, rather than a solved problem. They do not claim to have solved the general log-concavity for all $j$ or all strata, but rather to have established the first non-trivial cases and a framework for future investigation.