Prove the full $q$-deformed hit conjecture

Prove that, for every complex parameter $\alpha$ not of the form $-r/s$ with $r\in\{1,\ldots,n\}$, $s\in\mathbb N$, and $r\le s$, the quotient of $\mathbb C[z_1,\ldots,z_n]$ by the joint images of $D_{\alpha,k}=\sum_i z_i^k+\alpha\sum_i z_i^{k+1}\partial_i$ for $k\ge1$ is isomorphic to the graded $S_n$-coinvariant ring.

Background

Hivert and Thiéry proposed a deformation of the rational hit problem whose quotient is expected to agree with the coinvariant ring of the symmetric group for all nonsingular parameters. The paper proves only a slight weakening of this assertion, showing nonsingularity outside a larger exceptional set.

The unresolved part is to establish the conjecture at the parameters excluded by the paper’s weaker theorem and, in particular, to determine the precise singular set predicted by the original formulation.

References

In , Hivert and Thiéry made the following more general conjecture. Consider the quotient of the vector space of polynomials $\mathbb{C}[z_1,\ldots,z_n]$ by the joint image of the differential operators D_{\alpha,k}:=\sum_{i=1}nz_ik +\alpha\sum_{i=1}nz_i{k+1}\partial_i , \quad k\ge 1. For all $\alpha\in\mathbb{C}$ not of the form $-r/s$, where $r\in{1,\ldots,n}$ and $s\in\mathbb{N}$, $r\le s$, this vector space is isomorphic to the coinvariant ring of $S_n$ as a graded $S_n$-module.

— Twisted logarithmic de Rham complexes and the rational hit conjecture  (2610.26028 - Dotsenko, 19 Oct 2026) in Section 1.2, Conjecture 1 (the $q$-deformed hit conjecture)

To establish that the singular values belong to the range $r\le s$ predicted by the conjecture of Hivert and Thiéry, or to the more restrictive range $n\le s$ predicted by the conjecture of D'Adderio and Moci, a more precise analysis is needed.

— Twisted logarithmic de Rham complexes and the rational hit conjecture  (2610.26028 - Dotsenko, 19 Oct 2026) in Section 5.2, Remark following Theorem 5.5