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

翻译：我们研究对称多项式 $\prod_{α\in A_{n,d}}\bigl(1+α_1 x_1+\cdots+α_n x_n\bigr)$，其中 $A_{n,d}:=\{α\in\mathbb{Z}_{\ge 0}^n:|α|=d\}$，该多项式是 $\mathrm{Sym}^d(\mathbb{C}^n)$ 的全陈类，视为以 $α\in A_{n,d}$ 的权重 $α_1 x_1+\cdots+α_n x_n$ 为陈根的环面表示。其齐次 $k$ 次部分 $c_k(n,d)$ 是 $\mathrm{Sym}^d(\mathbb{C}^n)$ 的第 $k$ 个陈类。这些陈类及其在各对称函数基下的系数在枚举几何中处于核心地位。尽管定义简单，其系数的通用封闭公式却十分精细，且这些类的许多结构性质长期未得到充分理解。本文证明了关于其结构的若干猜想，建立了显式公式，并研究了陈类及其 $K$ 理论类比的对数凹性。在秩二情形下，通过过渡到舒尔基并将舒尔系数在 $d$ 的二项式基下展开，我们发现了一种新的二项式对数凹现象，并证明了精细的正性结果。本文展示了一种新颖的方法论：我们将多个AI系统与人类数学洞察在协同工作流中相结合，根据各工具在实验发现、猜想形成、符号证明构建和验证中的优势进行部署。据我们所知，这是首次详细阐述如何协调多种AI工具以在连贯的数学研究项目中取得实质性进展的案例研究之一。