On modular invariants of the truncated polynomial ring in rank four
Dang Vo Phuc
TL;DR
This work extends the Ha–Hai–Nghia program to rank four by conditional proof of the GL$_4$-invariant subspace of the truncated polynomial ring $\mathcal{Q}_m(4)$. It blends determinant calculus for the delta operator with Phuc's normalized Milnor derivations to derive rank-4 delta–Dickson identities and analyzes the Steenrod action on the Dickson algebra. A central contribution is the introduction and verification of the matching hypothesis $H_{\mathrm{match}}$, under which $\mathrm{Span}\,\mathcal{B}_m(4)$ becomes a $D_4$-submodule that generates $\mathcal{Q}_m(4)^{GL_4}$, and the Hilbert series matches the LRS polynomial $C_{4,m}(t)$. The results deliver a conditional confirmation of the Lewis–Reiner–Stanton conjecture at rank four and provide a structurally informed framework that unifies determinant methods with Steenrod-action techniques, supplemented by computational verification.
Abstract
We prove the rank-4 case of the conjecture of Ha-Hai-Nghia for the invariant subspace of the truncated polynomial ring $\mathcal{Q}_m(n)=\mathbb{F}_q[x_1,\dots,x_n]/(x_1^{q^m},\dots,x_n^{q^m}),$ under a new, explicit technical hypothesis. Our argument extends the determinant calculus for the delta operator by deriving crucial rank-4 identities governing its interaction with the Dickson algebra. We show that the proof of the conjecture reduces to a specific vanishing property, for which we introduce a sufficient condition, the "matching hypothesis" ($\mathrm{H_{match}}$), relating the degree structures of Dickson invariants. This condition is justified by theoretical arguments and verified computationally in many cases. Combining this approach with the normalized derivation approach from our prior work, we establish the conjecture. As a result, the Lewis-Reiner-Stanton Conjecture is also confirmed for rank four under the given hypothesis.