Coincidence of Reduced and Topological Zeta Invariants

Definitions and mathematical statement

Let $A$ be a $d$-dimensional algebra in the class intended by the source. The supplied record does not specify the base field or ring, the permitted algebra structures, or any restriction on $d$. It refers to a topological zeta function $Z_A^{\mathrm{top}}(s)$ and a reduced zeta function $Z_A^{\mathrm{red}}(T)$, but does not define these functions or specify what they count.

The symbols $m_{\mathrm{top}}(A)$ and $m_{\mathrm{red}}(A)$ denote invariants extracted, respectively, from the leading behavior of $Z_A^{\mathrm{top}}(s)$ at $s=0$ and of $Z_A^{\mathrm{red}}(T)$ at $T=1$. Their extraction formulas, including the powers, signs, and normalizations involved, are absent from the supplied record. Thus these descriptions do not determine the invariants precisely.

Coincidence conjecture, with unresolved definitions. For every algebra $A$ in the source's intended class, the conjectured assertion is \[ m_{\mathrm{red}}(A)=m_{\mathrm{top}}(A)\in{\mathbb{Q}}_{>0}. \] This is the supplied target, pending specification of the algebra class, the zeta functions, and the invariant normalizations; it is not a fully specified mathematical statement from the available evidence.

The excerpt additionally states that the topological degree conjecture is equivalent to $m_{\mathrm{top}}(A)$ being nonzero and finite. It also states that, if the Hilbert series conjecture holds, then $Z_A^{\mathrm{red}}(T)$ has a pole of order $d$ at $T=1$, whence $m_{\mathrm{red}}(A)$ is nonzero and finite. Neither auxiliary conjecture is formulated in the supplied record. These are source-reported relationships, not additional assumptions silently imposed on the coincidence conjecture.

Short English statement

The conjecture asserts that the invariants obtained from the topological and reduced zeta functions agree for every algebra covered by the source, and that their common value is a positive rational number. The supplied record omits the algebra hypotheses and the formulas defining these functions and invariants, so a fully precise formulation requires further source clarification.

Sources

• Local UnsolvedMath registry, supplied source record: permanent ID $30006435$, title Coincidence of Reduced and Topological Zeta Invariants, exact source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. The registry supplies the status open and the statement label corrected_verified; its verification note concerns formula delimiters, notation, grammar, and typography, rather than independent mathematical verification.
• Supplied report citation: Cohomology Theories for Automorphic Forms and Enumerative Algebra (2026), Oberwolfach Reports, DOI 10.4171/owr/2025/38, pp. 2065--2124. https://doi.org/10.4171/owr/2025/38. The registry records proposed year 2025 and names no proposer. The citation's 2026 title date and the 2025 provenance are retained as supplied.
• Supplied literature source: Oberwolfach Report 38. https://ems.press/content/serial-article-files/52251.
• Supplied literature source: Rossmann--Voll survey slides. https://torossmann.github.io/files/mfo25b.pdf. This supplied label does not establish attribution of the conjecture to the slide authors.
• Literature assessment dated exactly 2026-08-22: supplied dated curation, not independently verified. It labels the problem open and reports verification for selected families, computational support, and no known general proof. The original excerpt says that the conjecture had circulated informally for at least a decade and, to its unnamed author's knowledge at that time, had not previously been formally stated as such in a published document. It cites computations as [8, 9] and verified families as [13, §3.4]; those bibliography entries are absent from the supplied record.