Totally Positive Matrices and the Highest-Order Coefficients of the Characteristic Polynomial

Identifiability of tensor decompositions is often established through linear-algebraic conditions on the factor families. For nonnegative decompositions, however, positivity provides additional information that is not captured by dimension and independence alone: nonnegative terms cannot cancel, and their supports constrain competing decompositions. We introduce a positive scattering term that quantifies this additional source of identifiability and combine it with the dimension budget underlying the Lovitz--Petrov generalization of Kruskal's theorem. For every subset of components, we obtain two sufficient conditions: a threshold of $2|S|-2$ guarantees minimality and nonnegative rank, while the stronger threshold $2|S|-1$ guarantees uniqueness among nonnegative decompositions of the same length. The key result is a positive splitting inequality for irreducible exchanges of nonnegative rank-one tensors, which combines the dimension constraint with support-induced geometric rigidity. Although the scattering term is defined through an optimization over intermediate factor spaces, we show that its mode costs are exactly $0$, $1$, or $+\infty$, yielding an exact activation characterization in terms of graph connectivity. The resulting criterion can strictly certify sparse nonnegative tensor decompositions beyond the reach of Kruskal and Lovitz--Petrov conditions, including examples for which those conditions fail even after reshaping. In the matrix case, the two criteria reduce respectively to full-rank factorization and two-sided separability.

This work studies neural architectures for classifying symmetric positive-definite matrices, focusing on congruence-like layers, in which the input matrix is multiplied on the left and right by a (possibly rectangular) weight matrix $W$ and its transpose. Such layers lie at the core of the celebrated SPDNet and have also been employed independently for dimensionality reduction on positive-definite data. We show that the (semi)-orthogonality constraint commonly imposed on $W$ limits the expressivity of these layers: for certain activation functions, the resulting architecture collapses to a one-hidden-layer equivalent. This lack of expressivity follows from a loss of spectral diversity in congruence-like layers for semi-orthogonal $W$ and is a direct consequence of Poincaré's separation theorem. We then examine the choice of the final classifier, comparing several Riemannian classifiers and discussing their compatibility with the feature maps produced by congruence-like layers.

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.