An arXiv preprint constructs one-dimensional hypersurface families that land exactly on a general upper bound for generalized Loewy length, while other families fall one unit below it. The result turns an abstract question about reduction numbers into two precise outcomes. The paper studies reduction numbers for powers of a maximal ideal through a chosen generator, called a witness, and works with explicitly constructed rings. Its conclusions are algebraic existence results for the families under study.
The basic comparison is simple to state in words. If d is the order used for the maximal-ideal power and n is the relevant reduction number, the generalized Loewy length cannot exceed d times one more than n. The difference between that upper bound and the generalized Loewy length is the gap tracked in the paper. Its constructions produce a gap of zero and a gap of one, meaning the bound is reached exactly in one case and sits one unit above the generalized Loewy length in the other.
Two reference points
The setting is narrow by design: one-dimensional Cohen-Macaulay local rings and hypersurfaces, including examples over finite fields under stated algebraic conditions. The witness order is at least one. For hypersurfaces that meet a stated regular-initial-form condition, the generalized Loewy length lies between the Hilbert-Samuel multiplicity and that multiplicity plus the witness order minus one. These comparisons let the paper examine whether the reduction-based gap moves with the difference between generalized Loewy length and multiplicity.
In one family, z is formed as x plus twice y and serves as both a witness and a minimal reduction of the maximal ideal. The multiplicity and generalized Loewy length are each n plus two, while the reduction number is n plus one.
When the families diverge
Other constructions separate the two possible outcomes. In one family, z is formed as x squared plus xy plus y squared and is a witness and minimal reduction of the square of the maximal ideal. Its reduction number is the quantity p plus one divided by two, and the displayed relation reaches the general bound. In a related family using the same witness and reduction of the squared maximal ideal, the reduction number is p plus one and the generalized Loewy length is one below the corresponding bound.
For primes greater than two, a quadratic-nonresidue family uses z formed as x squared minus n times y squared, where n is a positive quadratic non-residue, meaning it is not a square in the relevant field. The element is a witness and minimal reduction of the squared maximal ideal. Its reduction number is p plus one divided by two, and the displayed relation leaves a one-unit gap.
A different family makes the multiplicity comparison explicit: its generalized Loewy length is exactly one greater than its multiplicity. Alongside the gap-zero and gap-one constructions, this supports the paper's statement that the reduction-number gap can vary independently of the difference between generalized Loewy length and multiplicity.
The result across prime characteristics
The most sweeping result is a pair of existence statements. For every prime p, the paper gives a one-dimensional hypersurface over the field Fp with an order-d witness for which the gap is zero, and another with an order-d witness for which the gap is one. In the zero-gap case, the witness order is at least one. The result shows that exact equality and a one-unit shortfall both occur in every prime characteristic covered by these constructions.
An additional family in Section 3 uses z formed as x squared plus xy plus y squared as a witness and minimal reduction of the squared maximal ideal. The reported reduction number is two n minus one, together with a corresponding equality involving generalized Loewy length. This adds another explicit equality to the paper's family-specific constructions.
Because the examples are stated for particular rings, fields and algebraic conditions, the findings should be read as constructions under stated hypotheses, not as a universal rule for all local rings. The manuscript is identified as an arXiv preprint, version 2 dated 3 Sep 2026.
Paper data and sources
Original title: Reduction numbers for witnesses to the generalized Loewy length
Authors: Richard Bartels, Sarah Dajani, Gabriel Koomson
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
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