Non-degeneracy conjecture for connected regular matroids

Equivalent formulations 2

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

1. The non-degeneracy conjecture for connected regular matroids

   A matroid $M$ of rank $r$ is degenerate if the degree of its Kazhdan–Lusztig polynomial $P_M(t)$ is strictly less than

   $$\left\lfloor \frac{r-1}{2} \right\rfloor.$$

   Non-degeneracy conjecture. Every connected regular matroid is non-degenerate. This conjecture concerns the degree of the Kazhdan–Lusztig polynomial for regular matroids; the source states that it is proved for modular matroids, while the general case remains open.

   source: Lorenzo Vecchi, “On matroid modularity and the coefficients of the inverse Kazhdan-Lusztig polynomial of a matroid”, arXiv:2103.08580 (2021).

2. Non-degeneracy conjecture for connected regular matroids

   Let $\mathsf{M}$ be a matroid. It is connected if every pair of distinct ground-set elements lies in a common circuit, and it is regular if it is representable over every field. It is non-degenerate when its Kazhdan–Lusztig polynomial $P_\mathsf{M}(t)$ has degree

   $$\left\lfloor\frac{\operatorname{rk}(\mathsf{M})-1}{2}\right\rfloor.$$

   Gedeon's non-degeneracy conjecture. Every connected regular matroid is non-degenerate. The conjecture remains open; the source emphasizes that regular matroids form a highly restrictive class, while the size of the representable-matroid class is asymptotically negligible.

   source: Luis Ferroni, George D. Nasr and Lorenzo Vecchi, “Stressed hyperplanes and Kazhdan-Lusztig gamma-positivity for matroids”, arXiv:2110.08869 (2022).