A mathematical preprint studies reduction numbers of powers m^d with respect to witnesses to generalized Loewy length across several infinite families of hypersurfaces arising from one-dimensional Cohen–Macaulay local rings. It reports families in which the difference d(r_z(m^d)+1) − gℓℓ(R) is 0 or 1, and says this difference can vary independently of gℓℓ(R) − e(R), where e(R) denotes Hilbert–Samuel multiplicity.
A bound built from one element
The main bound is conditional. If z belongs to m^d but not m^(d+1), reduces m^d, and has reduction number n, the proof uses the containments m^((n+1)d) ⊂ z m^(nd) ⊂ zR to obtain gℓℓ(R) ≤ d(n+1).
In ordinary terms, the reduction number records how many powers are needed before a selected element controls the next relevant power of the maximal ideal. That element is the witness. Under the stated assumptions, the inequality turns the witness’s reduction number into an upper bound for generalized Loewy length.
The paper states a corresponding bound for every principal reduction w of m^d: gℓℓ(R) ≤ d(r_w(m^d)+1). A principal reduction is a reduction generated by one element. With an infinite residue field, every minimal reduction of an m-primary ideal is principal, and r(I) equals r_w(I) for every principal reduction.
Exact formulas in selected families
One family has e(R) = gℓℓ(R) = n+2, while the witness reduction number is r_z(m) = n+1. Every principal reduction has that same number, and over an infinite field it also equals r(m). In this family, the generalized Loewy length and multiplicity coincide, while the witness reduction number is one lower.
In the characteristic-p power family, the paper gives p^n+2 = e(R) = gℓℓ(R) = r_z(m)+1, so r_z(m) = p^n+1. The same value holds for every principal reduction and, over an infinite field, for r(m).
A two-parameter family has gℓℓ(R) = e(R)+1 = 2^n p^m+3, and z = x^2+xy+y^2 is a witness. These are exact formulas for the stated family, under its prime and primitive-root assumptions.
How the gap changes
In one x^2+xy+y^2 family, z reduces m^2, gℓℓ(R) = p+3, and r_z(m^2) = (p+1)/2. Those values make 2(r_z(m^2)+1) − gℓℓ(R) = 0.
Another x^2+xy+y^2 family gives the one-unit gap: 2(p+2) = gℓℓ(R)+1 = 2(r_z(m^2)+1), with r_z(m^2) = p+1. The scaled reduction expression therefore exceeds generalized Loewy length by exactly 1.
Two further constructions extend the examples. In a finite-field construction based on a quadratic non-residue, z = x^2−ay^2 is a witness and minimal reduction of m^2. It reports p+3 = gℓℓ(R)+1 = 2(r_z(m^2)+1), with r_z(m^2) = (p+1)/2. In a binary hypersurface family with relation xy(x+y)^(2^n−1), z = x^2+xy+y^2 is a witness and minimal reduction of m^2, with r_z(m^2) = 2^(n−1) and gℓℓ(R) = 2(2^(n−1)+1).
The scope remains narrow
These formulas apply to selected algebraically defined infinite families, not to every one-dimensional Cohen–Macaulay local ring. Their constructions use stated field, characteristic, prime, primitive-root, and congruence conditions; conclusions about r(m) require an infinite residue field.
The document identifies itself as arXiv:2608.26003v1 [math.AC], dated 26 Aug 2026. R.B. and G.K. acknowledge financial support from the Trinity College Faculty Research Committee.
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
Original paper · Full text