Charalambous–Reeves Standard-Matching Conjecture

Definitions and mathematical statement

Attributed standard-matching conjecture. Let $k$ be a field, $S=k[x_1,\ldots,x_n]$, $\mathfrak a\subset S$ a monomial ideal with minimal monomial generating set $G(\mathfrak a)$, and $A=S/\mathfrak a$. Suppose $M$ is a standard matching on the Taylor resolution. The source asks whether the multigraded Poincaré series is\[P_A(\mathbf x,t)\stackrel{?}{=}\frac{\prod_{i=1}^n(1+tx_i)}{1+\sum_{I\in\mathcal U(M)}(-1)^{\operatorname{cl}(I)}m_I\,t^{\operatorname{cl}(I)+|I|}},\] where the summation is over subsets of generators left unmatched by $M$, $m_I=\operatorname{lcm}\{u:u\in I\}$, and $\operatorname{cl}(I)$ is the number of connected components of the graph with vertex set $I$ and an edge between distinct $u,v$ when $\gcd(u,v)\ne1$. Monomials $m_I$ are read as monomials in the formal multigrading variables $\mathbf x$, and $|I|$ is the number of generators in $I$.

Under the usual augmented graded-algebra convention, $P_A(\mathbf x,t)=\sum_{j\ge0}\sum_{\alpha\in{\mathbb{N}}^n}\dim_k\operatorname{Tor}^A_j(k,k)_\alpha\,\mathbf x^\alpha t^j$, where $k$ is the residue field at the homogeneous maximal ideal. This convention requires a proper positive-degree monomial ideal, and use of the numerator with all $n$ variables also needs the intended minimal-presentation hypotheses checked, especially if $\mathfrak a$ contains linear generators. A frequently relevant restriction is $\mathfrak a\subseteq(x_1,\ldots,x_n)^2$, but it is not stated in the record and is not silently added as a source hypothesis.

The definition of standard matching, including its additional conditions and which cells survive, is not supplied. Taylor cells are indexed by subsets of $G(\mathfrak a)$, but this does not define the special matching class. The empty-cell convention also matters: if $\varnothing$ were included in the displayed sum with $m_\varnothing=1$ and $\operatorname{cl}(\varnothing)=0$, the denominator would have constant term $2$, inconsistent with the usual constant term $1$ of $P_A$. An exclusion of the empty subset or another explicitly justified reduced convention is therefore needed; the notation $\mathcal U(M)$ in the displayed source formula is retained subject to this unresolved qualification. It cannot simply mean every unmatched subset under arbitrary conventions.

The original prose attributes the conjecture to Charalambous--Reeve and refers to the standard-matching product formula; the title instead spells Charalambous--Reeves. The dated literature triage checked 2026-08-21 labels the record open but explicitly says its extraction is incomplete and openness cannot be verified. The later curated field supplies the formula and carries a corrected_verified label reporting source inspection; these layers do not supply the missing matching definition or reconcile the dated warning.

Short English statement

The conjecture proposes a rational formula for the multigraded residue-field Poincaré series using unmatched Taylor cells. The special matching conditions, empty-cell convention, and minimal-presentation assumptions are missing, so the displayed formula is retained with those qualifications rather than asserted for arbitrary matchings.

Sources

• Local UnsolvedMath registry, permanent ID 30000081, code catalogue entry; title Charalambous--Reeves Standard-Matching Conjecture, while original prose says Charalambous--Reeve; Oberwolfach Reports Open Problems.
• Combinatorial Commutative Algebra (2005), Oberwolfach Reports, DOI 10.4171/owr/2004/32, pp. 1703--1746. https://doi.org/10.4171/owr/2004/32. Proposed year and year tag: 2004; no separate proposer field supplied.
• Supplied literature label Original Oberwolfach report: https://doi.org/10.4171/owr/2004/32. No separate authoritative conjecture citation or definition of standard matching is supplied.
• Literature triage checked 2026-08-21, status open, expressly qualified as not verifiable from its fragment; record updated 2026-09-16. The corrected_verified field reports checking the official report, but that check is not independently verified here.