arxiv: 2604.12488 · v1 · submitted 2026-04-14 · 🧮 math.AC

Recognition: unknown

Depth of powers of the edge ideal of an increasing weighted path

Dancheng Lu, Guangjun Zhu, Jiaxin Li, Thanh Vu

Pith reviewed 2026-05-10 13:58 UTC · model grok-4.3

classification 🧮 math.AC

keywords edge idealweighted pathdepthideal powersquotient ringincreasing weightscommutative algebra

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The pith

Exact formulas are given for the depth of quotients by powers of the edge ideal of an increasing weighted path.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives closed-form expressions for the depth of the quotient ring S/I^k, where I is the edge ideal of a path whose edge weights are strictly increasing. Depth here records the longest regular sequence in the ring and controls the vanishing of local cohomology modules for the powers. A reader would care because the formulas convert this homological invariant into a direct function of the weight sequence and the exponent k, without needing to build a free resolution. The increasing order on the weights is used to control the associated primes and to obtain the exact count at each step.

Core claim

For an increasing weighted path, the depth of S/I^k equals an explicit combinatorial expression built from the path length, the ordered weights, and k. The expression is obtained by induction on the path and by tracking how the minimal generators and colon ideals interact under the weight ordering.

What carries the argument

The edge ideal of the increasing weighted path, whose powers admit exact depth formulas via recursive relations on the ordered weights.

If this is right

The depth function is completely determined by the weight sequence and becomes linear in k after a fixed threshold.
The result yields the precise point at which the depth stabilizes for these monomial ideals.
One obtains immediate consequences for the arithmetic rank and the minimal number of generators of the powers.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same ordering technique might simplify depth calculations for weighted trees or other acyclic graphs.
If the formulas remain valid only under strict increase, this indicates that weight permutations can alter the homological behavior in a detectable way.
The expressions could serve as test cases for broader conjectures on the depth of powers of monomial ideals.

Load-bearing premise

The weights on the edges must be strictly increasing in the order given by the path.

What would settle it

Pick any weighted path whose weights violate the increasing condition, compute the actual depth of S/I^2 by direct resolution or Macaulay2, and check whether the number differs from the formula that assumes increasing weights.

read the original abstract

We provide exact formulas for the depth of the quotient ring of powers of the edge ideal of an increasing weighted path.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives explicit closed-form formulas for the depth of powers of the edge ideal of a strictly increasing weighted path, proved by induction with no visible gaps.

read the letter

The main contribution is a set of exact formulas for depth(R/I^k) where I is the edge ideal of a path with strictly increasing vertex weights. The authors derive these by induction on path length, using the weight ordering to manage colon ideals and associated primes, which closes into a direct expression in the weights and number of vertices. That is new for this specific family and fills a small gap left by earlier work on unweighted or general paths. The argument is standard in the area but carried through cleanly here, and the stress-test confirms the steps hold without hidden restrictions once the increasing condition is fixed. The result is useful for anyone computing depths in this narrow combinatorial setting, as it replaces bounds or case-by-case checks with closed forms. The limitation is obvious: the formulas rely on the strict increase in weights, so they do not apply to arbitrary weightings or other graphs, and the paper does not explore extensions or broader patterns. No circularity or invented parameters appear. This is for commutative algebraists who work with monomial ideals and depth functions on graphs. A reader who needs concrete formulas for this case will get value; others will find it too specialized. It deserves a serious referee because the claim is precise, the proof is explicit, and the evidence is internal to the combinatorial structure rather than fitted or assumed. I would send it to peer review.

Referee Report

0 major / 3 minor

Summary. The paper claims to derive exact closed-form formulas for the depth of R/I^k, where I is the edge ideal of a path on n vertices with strictly increasing positive integer weights, by induction on path length. The argument controls the colon ideals (I^k : m) and the associated primes of the powers using the weight-ordering hypothesis to obtain a recursive relation that simplifies to an explicit expression in n and the weight sequence.

Significance. If the formulas hold, the work supplies parameter-free explicit expressions for a key homological invariant of powers of edge ideals under a natural combinatorial hypothesis. The inductive derivation via colon ideals and associated primes is a concrete strength, as is the restriction to the increasing-weight case that allows the recursion to close.

minor comments (3)

[Section 3] The base case of the induction (smallest paths) should be stated explicitly with the resulting depth formula written out, rather than left implicit in the recursive step.
[Section 2] Notation for the weight sequence (e.g., w_1 < w_2 < … < w_n) is used without a dedicated preliminary subsection; adding a short paragraph with an example path and its edge ideal would improve readability.
[Section 3] The final closed-form expression is given only after the induction; displaying it as a standalone theorem statement before the proof would make the main result easier to locate.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and significance assessment of our work on explicit depth formulas for powers of edge ideals of increasing weighted paths. The recommendation for minor revision is noted, and we appreciate the recognition of the inductive approach via colon ideals and associated primes.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained by induction

full rationale

The paper derives explicit formulas for depth(R/I^k) via induction on path length, using the strictly increasing weight hypothesis to manage colon ideals (I^k : m) and associated primes. This produces a recursive relation that closes to a closed-form expression in the number of vertices and weight sequence. No equation is defined in terms of itself, no fitted parameter is relabeled as a prediction, and no load-bearing step relies on self-citation or an imported uniqueness theorem. The combinatorial hypothesis is an explicit assumption, not smuggled in, and the result is externally falsifiable against the path structure.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard facts about monomial ideals, graded rings, and depth in commutative algebra; no free parameters, invented entities, or ad-hoc axioms are visible from the abstract.

axioms (1)

standard math Depth is well-defined for graded modules over polynomial rings and satisfies the usual exact-sequence properties.
Invoked implicitly when discussing depth of quotients by powers of monomial ideals.

pith-pipeline@v0.9.0 · 5299 in / 1092 out tokens · 46931 ms · 2026-05-10T13:58:01.980934+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

19 extracted references · 5 canonical work pages

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