arXiv subjects

Ronnie Cheng

Publications and source records attributed to Ronnie Cheng.

6 recordsLinked to original sources

arXiv2026

Kazhdan-Lusztig polynomials of matroids need not be unimodal

We construct, over every finite field, representable matroids whose Kazhdan-Lusztig polynomials are not unimodal. In particular, the conjectures that all Kazhdan-Lusztig polynomials of matroids are log-concave and that they are real-rooted are both false. Our examples are obtained by deleting points from finite projective geometries. More generally, we prove that, under a half-rank degree condition in each contraction quotient, the Kazhdan-Lusztig polynomial of every contraction enumerates the subspaces whose projective points lie in the corresponding deleted set, while the $Z$-polynomial agrees with that of the full projective geometry.

math.CO↗

arXiv2026

Tangent classes of matroids and wonderful compactifications

For every loopless matroid $M$ and every Feichtner--Yuzvinsky building set $\mathcal{G}$ containing the top flat, we construct an integral tangent class $T_{M,\mathcal{G}}^{\mathbb{Z}}\in K_{\mathbb{Z}}(M,\mathcal{G})$; in the realizable case it specializes to the class of the tangent bundle of the corresponding wonderful compactification, it recovers the Hilbert series of the Chow ring through Hirzebruch--Riemann--Roch, and it satisfies the expected Chern-alpha lower bounds. This reproduces the tangent class and its key properties studied by the first author in arXiv:2606.22650. The main body of this paper was produced autonomously, without human mathematical guidance, by Danus, an AI mathematical reasoning agent. Danus solved the problem before arXiv:2606.22650 was publicly available, demonstrating the potential of AI agents in mathematical research. We reproduce its output faithfully, adding only editorial comments; the experiment is documented in Appendix B.

math.AG↗

arXiv2026

Danus: Orchestrating Mathematical Reasoning Agents with Fact-Graph Memory

Recent LLM-based mathematical reasoning agents have begun to tackle research-level problems and, in several cases, have contributed to the resolution of open problems. However, scaling and orchestrating such agents effectively remains challenging, due to the difficulty of coordinating parallel proof search while keeping intermediate claims organized and reliable. In this paper, we propose Danus, an orchestration system for research-level mathematical reasoning centered on a shared fact graph as a global memory-management mechanism. Danus consists of a main agent that performs planning and coordination, multiple worker agents that carry out proof search in parallel, and a stateless verifier that checks proposed mathematical claims before they are admitted into the fact graph. Each verified fact is stored together with its proof and logical dependencies, allowing the system to build long arguments incrementally while keeping the shared proof state organized. The main agent periodically summarizes the evolving proof state, redirects workers across promising directions, and supports interaction with human mathematicians through progress reports. We evaluate Danus through six research-level case studies in algebraic geometry, singularity theory, and combinatorics, illustrating how the fact-graph memory mechanism enables Danus to construct long, detailed mathematical proofs. Our results suggest that fact-graph-based orchestration provides an effective route toward scaling mathematical reasoning agents for long-horizon research problems. Danus is open source at https://github.com/frenzymath/Danus.

cs.AI↗

arXiv2026

Tangent classes for matroid building sets

Significant combinatorial constraints and structure on matroids can come from notions in algebraic geometry, even without the matroids themselves being representable. Let $M$ be a loopless matroid on a finite ground set $E$, and let $\G$ be a building set containing the top flat $E$. We define a tangent class $T_{M,\G}$ in the $K$-ring $K(M,\G)$, which extends the tangent bundle class of the de Concini--Procesi wonderful model from realizable matroids to arbitrary matroids with building sets. The class $T_{M,\G}$ satisfies a matroidal Hirzebruch--Riemann--Roch package. More precisely, its Hirzebruch class specializes to the Todd class and computes the Chow polynomial of $(M,\G)$. In the realizable case, these identities agree with the usual tangent-bundle computations on the corresponding wonderful model. As an application, we prove Chern-number inequalities for $T_{M,\G}$, including a Miyaoka--Yau type inequality with respect to the hyperplane class.

math.AG↗

arXiv2026

Inequalities for Chow Polynomials and Chern Numbers of Matroids

The Chow polynomial of a matroid is a fundamental invariant whose coefficients exhibit strong positivity properties, including $\gamma$-positivity. We interpret the normalized Chow coefficients as a probability distribution and establish new inequalities for its central moments. As consequences, we obtain bounds on the number of flags of flats and inequalities on the roots of the Chow polynomial. We further relate these moment inequalities to algebraic geometry via the Hirzebruch $\chi_y$-genus. This yields new inequalities for matroidal Chern numbers. In particular, for any matroid of rank $d+1$, we prove that $c_1c_{d-1}\le c_d$, with equality if and only if $d=1$ or the simplification of the matroid is Boolean.

math.CO↗

arXiv2025

On the tangent bundle and the divisor theory of a general matroid

Extending classical algebro-geometric constructions to arbitrary matroids, we construct a $K$-class $T_M\in K(M)$ for every loopless matroid $M$. When $M$ is realizable by a linear subspace $L$, $T_M$ recovers the $K$-class of the tangent bundle of the wonderful compactification $W_L$. We derive two formulas for the total Chern class of $T_M$ (one combinatorial and one geometric) and show that the associated Todd class agrees with the Todd class appearing in the matroid Hirzebruch--Riemann--Roch formula. To develop a positivity theory entirely at the combinatorial level, we introduce the notion of ``fake effective cone,'' a combinatorial analogue of the classical effective cone, and use it to characterize big and nef divisors in $A(M)$. Finally, we define the $\beta_S$ classes, obtained from Cremona conjugates of the classical $\alpha_S$ classes, and study their properties to provide a rich and computable family of combinatorially nef divisors.

math.AG↗