Rossmann's simple-pole conjecture for nilpotent algebra zeta functions

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Let O\mathfrak{O} be the coefficient ring, let AA be an nn-dimensional nilpotent O\mathfrak{O}-algebra, and write ζA(t)\zeta_A(t) and ζA⊲(t)\zeta^{\lhd}_A(t) for its subalgebra and ideal zeta functions. Let (∞n−1)q=1(1−q)…(1−qn−1)\binom{\infty}{n-1}_q=\frac{1}{(1-q)\dots(1-q^{n-1})}. Rossmann's simple-pole conjecture. Both ζA(t)\zeta_A(t) and ζA⊲(t)\zeta^{\lhd}_A(t) have a simple pole at t=1t=1, with residue (∞n−1)q\binom{\infty}{n-1}_q. This refines the pole-existence conjecture for nilpotent algebras; the supplied text notes that the additional nilpotence requirement is essential and gives no indication that the conjecture itself has been resolved.

References

Primary source

Tomas Reunbrouck, “p-Adic Asymptotic Subalgebra Enumeration”, arXiv:2605.20422 (2026).

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