Invariant Rings of Adjacent-Quadratic Triangular Derivations
Abstract
Let k be a field of characteristic zero and DN(an)=an+1an+2 the adjacent-quadratic locally nilpotent derivation on AN=k[a0,…,aN]. We determine the invariant ring at N=5, where deleting a0 gives the polynomial base kernel BD5=K=k[p,q,J1,J2], so localization gives the polynomial-line invariant algebra (A5D5)pq=Kpq[W] with W=F0/(p2q4), but this does not by itself determine the global kernel. An invariant H of weight $24$ and a0-degree $2$ lies outside K[F0], and adjoining it gives the six-generator hypersurface C5; the obstruction is explained by algebraic residue behavior defeating naive denominator descent. An invariant F46int of weight $46$ and a0-degree $4$ lies outside C5 and satisfies p2F46int∈ C5. To pass from this candidate to the full invariant ring, a second slice chart arising from f13, together with exact q- and p-saturation, yields A5D5=C5[F46int], hence finite generation, with a codimension-two complete-intersection presentation and explicit Hilbert series. At N=6 the transported B-level ring is exact, while the A6 analysis is established through total degree $13$ and the full A6 invariant ring remains open.
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BibTeX
@misc{adly2026invariant,
title = {{Invariant Rings of Adjacent-Quadratic Triangular Derivations}},
author = {Ahmed M. Adly},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02604},
eprint = {2610.02604},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02604},
}
RIS
TY - UNPB
TI - Invariant Rings of Adjacent-Quadratic Triangular Derivations
AU - Ahmed M. Adly
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.02604
UR - https://arxiv.org/abs/2610.02604
AB - Let k be a field of characteristic zero and D_N(a_n)=a_n+1a_n+2 the adjacent-quadratic locally nilpotent derivation on A_N=k[a_0,…,a_N]. We determine the invariant ring at N=5, where deleting a_0 gives the polynomial base kernel B^D_5=K=k[p,q,J_1,J_2], so localization gives the polynomial-line invariant algebra (A_5^D_5)_pq=K_pq[W] with W=F_0/(p^2q^4), but this does not by itself determine the global kernel. An invariant H of weight $24$ and a_0-degree $2$ lies outside K[F_0], and adjoining it gives the six-generator hypersurface C_5; the obstruction is explained by algebraic residue behavior defeating naive denominator descent. An invariant F_46^int of weight $46$ and a_0-degree $4$ lies outside C_5 and satisfies p^2F_46^int∈ C_5. To pass from this candidate to the full invariant ring, a second slice chart arising from f_13, together with exact q- and p-saturation, yields A_5^D_5=C_5[F_46^int], hence finite generation, with a codimension-two complete-intersection presentation and explicit Hilbert series. At N=6 the transported B-level ring is exact, while the A_6 analysis is established through total degree $13$ and the full A_6 invariant ring remains open.
ER -