A mathematical preprint reports that the associated-prime set of a monomial ideal can change as the ideal is raised to successive powers. In its first displayed fluctuating example, the maximal ideal is absent from powers 1 and 2, present at powers 3, 4 and 5, and absent again at powers 6 and 7. A broader theorem states that infinitely many monomial ideals show fluctuation in every dimension at least three.
For a general reader, this is a membership question inside an algebraic record: at each power, the authors ask whether a designated object, the maximal ideal, belongs to the set of prime ideals associated with that power. The paper studies monomial ideals and their powers in polynomial rings over a field. Its main criteria are formulated in the three-variable ring K[x, y, z], alongside constructions in higher dimensions.
A test based on exponents
To answer that question, the authors combine depth-theoretic and combinatorial methods. Depth is an algebraic quantity used in the paper's reasoning, while the combinatorial part examines how the generators' exponents are ordered. Proposition 3.3 says that the displayed exponent-order conditions are sufficient to make the maximal ideal associated. It also identifies a corner element and the corresponding irreducible component.
The companion Proposition 3.4 supplies the opposite test. When generator exponents in two distinct variables are nonincreasing, it states that the maximal ideal is not associated. Taken together, the two criteria make exponent order a guide to whether the maximal ideal appears in the associated-prime set for a given power.
From thresholds to reversals
Two threshold constructions show opposite one-way patterns. In Example 4.5, for any positive integer m, the maximal ideal is associated for powers 1 through m and absent from power m+1 onward. Example 4.6 reverses that behavior: the maximal ideal is absent for powers 1 through m and associated from power m+1 onward.
Example 5.1 breaks even the threshold picture. Its first displayed ideal has the maximal ideal out at powers 1 and 2, in at powers 3 through 5, and out again at powers 6 and 7. The membership does not simply stay fixed after the first change, which is the fluctuation the paper places at the center of its analysis.
A pattern that scales
The theorem-level result broadens the point beyond individual examples. Theorem 5.8 states that for every dimension at least three, infinitely many monomial ideals have fluctuation in the associated primes of their powers. The claim covers every stated dimension from three upward, so the paper presents fluctuation as a repeatable phenomenon within the class it studies.
The preprint also reports negative answers to two related structural properties. Proposition 5.14 states that, in every dimension at least three, infinitely many nearly normally torsion-free monomial ideals fail persistence. Proposition 5.20 gives the matching result for copersistence, again with infinitely many co-nearly normally torsion-free monomial ideals in every dimension at least three. These constructions show that the associated-prime sets in the studied families need not obey the proposed one-way rules in either direction.
The square-free boundary
A separate boundary appears for square-free monomial ideals, a narrower class considered in the conclusion. The paper states that no such ideal fluctuates in dimensions up to five. Using expansion, it says counterexamples can be constructed in dimensions seven and above. Dimension six remains open, leaving the six-variable case as the unresolved point between those results.
Another displayed construction answers a proposed invariance question. In the three-variable ring K[x, y, z], the maximal ideal is present for the original ideal but absent for its second power. That example disproves the idea that the associated-prime set must be unchanged across powers, even at the first step.
What the result does not settle
The study is theoretical and analyzes constructed monomial ideals and their powers rather than an empirical sample. Its conclusions concern monomial ideals and do not directly establish the same behavior for arbitrary non-monomial ideals. The examples also do not show that the constructed ideals are representative of all monomial ideals.
The square-free question is still unresolved in six variables, and some reported examples rely on Macaulay2 calculations without computational details or reproduced output. The document is identified in its front matter as arXiv:2608.28243v1, dated 28 August 2026, and as a preprint.
Paper data and sources
Original title: On the existence of the maximal ideal in the set of associated primes of monomial ideals
Authors: M. Cimpoeaş, M. Nasernejad, A. A. Qureshi
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-28
DOI: Not available
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