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

Abstract: We study the symmetric polynomial $$\prod_{α\in A_{n,d}}\bigl(1+α<em>1 x_1+\cdots+α_n x_n\bigr)$$ where $$A</em>{n,d}:={α\in\mathbb{Z}<em>{\ge 0}<sup>n:|α|=d}$$, which is the total Chern class of $\mathrm{Sym}<sup>d(\mathbb{C}<sup>n)$, viewed as a torus representation whose Chern roots are the weights $α_1 x_1+\cdots+α_n x_n$ for $α\in A</em>{n,d}$. Its homogeneous degree-$k$ part $c_k(n,d)$ is the $k$-th Chern class of $\mathrm{Sym}<sup>d(\mathbb{C}<sup>n)$. 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.

• The paper proves separate polynomiality and universal binomial formulations for symmetric power Chern classes, resolving key conjectures.
• It integrates synchronized AI-assisted methods with human insight to achieve explicit structural results and combinatorial decompositions.
• The work uncovers new positivity and log-concavity phenomena in Schur expansions, impacting computational algebraic geometry and K-theory.

Positivity Structures and AI-Human Collaboration in Classical Enumerative Geometry

Introduction and Context

This paper investigates structural and positivity properties of Chern classes associated with symmetric powers of vector bundles, a central object in intersection theory, enumerative geometry, and representation theory. Starting from the total Chern class of the symmetric power $\Sym^d(E)$, where $E$ is a rank-$n$ bundle, the work expands these classes in various classical symmetric function bases and analyzes their combinatorial, algebraic, and geometric properties, focusing especially on their coefficients and positivity/log-concavity phenomena.

Notably, the methodology integrates a synchronized AI-assisted workflow, orchestrating the complementary capabilities of different machine learning agents along with directed human insight to achieve explicit structural results and resolve conjectures regarding polynomiality and log-concavity. This paper offers a detailed case study of how contemporary AI systems, including DeepMind’s AlphaEvolve, GPT-5.5 Pro, and Co-Mathematician, can be systematically coordinated to contribute significant advances on subtle, technically involved conjectures in algebraic geometry.

Polynomiality and Binomiality of Chern Classes

The Chern class of the symmetric power $\Sym^d(E)$, expressed as a symmetric polynomial in the Chern roots $x_1,\ldots,x_n$, admits expansion in bases such as elementary symmetric functions ($e_\lambda$) and Schur functions ($s_\lambda$). Denoting the $k$-th homogeneous part by $c_k(n,d)$, the central algebraic question concerns the structure and parameter dependence of the coefficients $f_\lambda(n,d)$ in these expansions: $E$0

The paper settles two precise conjectures:

1. Separate Polynomiality: For fixed $E$1 and partition $E$2, $E$3 is polynomial separately in $E$4 and $E$5.
2. Strong Binomiality: In fact, for each $E$6 and $E$7, $E$8 is a universal polynomial in a finite collection of binomial coefficients $E$9 ($n$0).

The proofs invoke explicit generating function computations, combinatorial decompositions via Newton identities, and reduction of power sum moments over weak compositions to polynomials in the distinguished binomials. The explicit closed forms for low-degree Chern classes (notably, the $n$1 formula) and the general polynomial structure for all $n$2 are rigorously established, with critical input from symbolic AI and computational searches.

Positivity and Log-Concavity: Rank-2 Chern-Schur Expansion

The study then focuses on refined positivity and log-concavity properties for the Schur expansion of Chern classes in the case $n$3. The Schur coefficients $n$4 in

$n$5

are further expanded in the binomial basis of $n$6: $n$7

Two main conjectures are proven in the first nontrivial cases by a blend of AI-generated conjecturing/proof drafting and human-guided induction:

• Binomial Positivity: For $n$8, $n$9 for all $\Sym^d(E)$0.
• Binomial Log-Concavity: For $\Sym^d(E)$1, the sequence $\Sym^d(E)$2 is log-concave.

The proofs synthesize explicit combinatorial models (involving defect-derangements and fixed-point-free permutations) with binomial basis recurrence relations. The AI systems contributed both formula discovery and the construction of recurrence relations, while humans supplied the crucial changes of basis and perspective, and conducted inductive arguments via binomial-basis manipulation and careful combinatorics.

A key result is that binomial log-concavity of the coefficient sequence for each $\Sym^d(E)$3 implies ordinary log-concavity of the value-sequence $\Sym^d(E)$4.

$\Sym^d(E)$5-Theoretic Extension

The work generalizes binomial polynomiality phenomena to the $\Sym^d(E)$6-theoretic setting via normalized equivariant multiplicative Chern classes: $\Sym^d(E)$7 Diagonal extraction of the coefficient $\Sym^d(E)$8 recovers $\Sym^d(E)$9, showing that the binomial polynomiality phenomenon extends to coefficients of all minors of $x_1,\ldots,x_n$0.

Plücker Strata and Generalizations

A further extension considers the equivariant cohomology classes of deeper coincident root strata (Plücker strata), and studies Schur coefficients of these strata as explicit polynomials in $x_1,\ldots,x_n$1. Upon a natural parameter shift, all observed cases exhibit positive and log-concave expansions in the shifted binomial basis. This leads to the conjecture that, for all contact partitions and indices, the shifted binomial coefficients in the Schur expansion are nonnegative and log-concave.

Although the recursive combinatorial structure for producing equivariant Plücker strata cohomology classes is available, a general positivity/log-concavity-preserving transition kernel for the shifted-basis recursion is not yet established. Moreover, standard Hodge-theoretic or real-rootedness arguments do not suffice due to the lack of intersection-theoretic realization of these shifted coefficients, and complex root structure in higher-degree cases.

Synchronized AI-Human Research Methodology

This work is notable for the explicit and transparent orchestration of multiple AI agents, each deployed according to their respective computational strengths. Roles included formula conjecturing, symbolic manipulation, recurrence detection, proof drafting, and verification. The human researchers provided problem decomposition, conjecture synthesis, and rigorous proof construction based on AI output. This ensemble approach enabled progress on structure theorems, closed formulas, and subtle log-concavity conjectures that were otherwise intractable by standard methods.

The division of labor is summarized as:

• AlphaEvolve: Experimental discovery of closed forms and recurrences.
• ChatGPT 5.5 Pro: Symbolic derivation, formula synthesis, and proof drafting.
• Co-Mathematician: Targeted proofs for recalcitrant binomial positivity/log-concavity claims and high-level induction schema formulation.

Implications and Future Directions

The explicit results provided (universal binomial formulas for symmetric power Chern classes, binomial-basis positivity/log-concavity results, and their $x_1,\ldots,x_n$2-theoretic extensions) enhance algorithmic intersection theory, the enumerative geometry of Fano schemes, and the combinatorics of characteristic classes. The log-concavity properties suggest the existence of deeper, as yet unclassified, positivity phenomena at the level of root and contact strata in projective geometry, and point toward geometric invariants with Lorentzian or mixed Hodge-theoretic underpinnings.

The AI-assisted mathematical workflow demonstrates that mechanized and data-driven search, when conducted in a transparent, modular fashion and tightly coupled with expert human abstraction, can achieve new structural results in contemporary mathematics—particularly for problems where combinatorial explosion and lack of direct combinatorial models prevent closed-form solution by traditional means.

Open directions include:

• Full proofs of posited binomial positivity and log-concavity for higher Schur modes.
• Deeper understanding and classification of the transition kernels underlying shifted-binomial recursions for Plücker strata.
• Extension of these phenomena to Chern–Schwartz–MacPherson classes and, more broadly, to characteristic classes of singularity loci and related geometric situations.

Conclusion

This work provides rigorous structure theorems and explicit positivity/log-concavity results for Chern classes of symmetric powers in classical enumerative geometry, pushing the boundaries of current knowledge concerning binomiality and basis expansion phenomena. The coordinated use of multiple AI systems, each serving specific roles tightly integrated with human insight, establishes a reproducible template for AI-human ensemble research in mathematics. The results represent a nontrivial advance in the algorithmic, combinatorial, and geometric understanding of symmetric powers, Schur expansions, and root-collision loci, with implications for future developments in symbolic mathematics, computational algebraic geometry, and the foundations of automated mathematical discovery.