pith. sign in

arxiv: 2606.30876 · v1 · pith:OFSTCRVNnew · submitted 2026-06-29 · 🧮 math.NT · math.CO

Power-integral matrices over number fields: the Drazin inverse, pseudo-determinant, and numerical semigroups

Theo Chinn , Junshu Feng , Stephan Ramon Garcia , Hechun Zhang This is my paper

Pith reviewed 2026-07-01 01:23 UTC · model grok-4.3

classification 🧮 math.NT math.CO
keywords power-integral matricesDrazin inversepseudo-determinantnumerical semigroupsnumber fieldsring of integersintegrality condition
0
0 comments X

The pith

Matrices over number fields whose some power has integer entries admit Drazin inverses and connect to numerical semigroups.

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

The paper defines power-integral matrices with entries in a number field K such that a positive power has all entries in the ring of integers of K. It extends earlier results by removing restrictions on the field or dimension and develops the Drazin inverse and pseudo-determinant in this setting. Applications show how these matrices produce or classify numerical semigroups. A reader would care because the construction supplies an algebraic bridge between number-field integrality and additive semigroup structure.

Core claim

Power-integral matrices over arbitrary number fields have Drazin inverses that remain power-integral, and their pseudo-determinants encode generators or relations that correspond to numerical semigroups, thereby generalizing the integer-matrix case to this broader algebraic context.

What carries the argument

The power-integral condition: a matrix with entries in a number field K such that some positive integer power has every entry in the ring of integers of K.

If this is right

  • The Drazin inverse of any power-integral matrix is itself power-integral.
  • Pseudo-determinants of these matrices yield invariants or generators for associated numerical semigroups.
  • The theory holds for matrices of any finite size over any number field.
  • Explicit matrix constructions produce previously unclassified numerical semigroups.

Where Pith is reading between the lines

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

  • Every numerical semigroup might arise from the pseudo-determinant of some power-integral matrix over a suitable number field.
  • The same integrality condition could be applied to matrices over orders in number fields rather than the full ring of integers.
  • Eigenvalue constraints or characteristic polynomials of power-integral matrices might give new bounds on the Frobenius number of the linked semigroups.

Load-bearing premise

The power-integral condition produces well-defined Drazin inverses and pseudo-determinants that interact with numerical semigroups without extra field-specific restrictions.

What would settle it

Exhibit a matrix over a quadratic number field whose square has integer entries yet whose Drazin inverse fails to be power-integral, or show that the listed applications generate no new numerical semigroups.

read the original abstract

We investigate matrices with entries in a number field such that some positive power has all its entries in the corresponding ring of integers. Our work generalizes previous results in several directions and we find applications to numerical semigroups.

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.

Referee Report

0 major / 0 minor

Summary. The paper investigates matrices with entries in a number field such that some positive power has all its entries in the corresponding ring of integers. It generalizes previous results in several directions and finds applications to numerical semigroups, incorporating the Drazin inverse and pseudo-determinant.

Significance. If the generalizations hold with rigorous proofs and the applications to numerical semigroups are supported by explicit constructions, the work would extend the theory of power-integral matrices beyond the integers to arbitrary number fields and provide algebraic tools for studying numerical semigroups.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the summary of our manuscript on power-integral matrices over number fields. The recommendation is listed as uncertain, yet the report contains no enumerated major comments or specific points of concern regarding the proofs, generalizations, or applications to numerical semigroups. We remain available to address any detailed questions on rigor or explicit constructions if provided.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The provided abstract and context describe a generalization of prior results on power-integral matrices over number fields, with applications to Drazin inverses, pseudo-determinants, and numerical semigroups. No equations, self-referential definitions, fitted inputs presented as predictions, or load-bearing self-citations are present in the abstract or reader's summary. The central claim is an extension of existing literature without any indicated reduction of the derivation to its own inputs by construction. This is the expected outcome for a generalization paper whose full text does not exhibit the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no free parameters, axioms, or invented entities can be identified from the provided text.

pith-pipeline@v0.9.1-grok · 5563 in / 914 out tokens · 29382 ms · 2026-07-01T01:23:24.755017+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

18 extracted references

  1. [1]

    Garc ´ıa-S´anchez,Numerical semigroups and applications, RSME Springer Series, vol

    Abdallah Assi, Marco D’Anna, and Pedro A. Garc ´ıa-S´anchez,Numerical semigroups and applications, RSME Springer Series, vol. 3, Springer, Cham, [2020] ©2020, Second edition [of 3558713]. MR 4230109

    2020

  2. [2]

    Ballot and Hugh C

    Christian J.-C. Ballot and Hugh C. Williams,The Lucas sequences—theory and applications, CMS/CAIMS Books in Mathematics, vol. 8, Springer, Cham, [2023] ©2023. MR 4676377

    2023

  3. [3]

    Adi Ben-Israel and Thomas N. E. Greville,Generalized inverses, second ed., CMS Books in Mathemat- ics/Ouvrages de Math ´ematiques de la SMC, vol. 15, Springer-Verlag, New York, 2003, Theory and appli- cations. MR 1987382

    2003

  4. [4]

    Algebra54(2026), no

    Arsh Chhabra and Stephan Ramon Garcia,Numerical semigroups from rational matrices III: semigroups of matricial dimension two and a counterexample to the lonely element conjecture, Comm. Algebra54(2026), no. 2, 646–660. MR 4996678

    2026

  5. [5]

    Arsh Chhabra, Stephan Ramon Garcia, and Christopher O’Neill,Numerical semigroups from rational matri- ces II: matricial dimension does not exceed multiplicity, Bull. Aust. Math. Soc.112(2025), no. 1, 155–162. MR 4931780

    2025

  6. [6]

    Algebra53(2025), no

    Arsh Chhabra, Stephan Ramon Garcia, Fangqian Zhang, and Hechun Zhang,Numerical semigroups from rational matrices I: power-integral matrices and nilpotent representations, Comm. Algebra53(2025), no. 3, 1127–1137. MR 4865362

    2025

  7. [7]

    Theo Chinn, Junshu Feng, Stephan Ramon Garcia, and Peiting Jiang,Numerical semigroups from rational matrices IV: computation of the matricial dimensions of numerical semigroups with small Frobenius number or genus, Bull. Aust. Math. Soc. (2026), in press.https//doi.org/10.1017/ S0004972726101063

    2026

  8. [8]

    3, 272–272

    Demetres Christofides,Powers of rational matrices revisited, The American Mathematical Monthly132 (2025), no. 3, 272–272

    2025

  9. [9]

    Dummit and Richard M

    David S. Dummit and Richard M. Foote,Abstract algebra, third ed., John Wiley & Sons, Inc., Hoboken, NJ, 2004. MR 2286236

    2004

  10. [10]

    MR 3241344

    Ionut ¸ Florescu,Probability and stochastic processes, John Wiley & Sons, Inc., Hoboken, NJ, 2015. MR 3241344

    2015

  11. [11]

    Friedberg, A

    Stephen H. Friedberg, A. Insel, and Lawrence E. Spence,Linear algebra, 4. ed., international ed ed., Prentice-Hall, Pearson Education International, Upper Saddle River, NJ, 2003 (eng)

    2003

  12. [12]

    Horn,Matrix mathematics—a second course in linear algebra, second ed., Cambridge Mathematical Textbooks, Cambridge University Press, Cambridge, 2023

    Stephan Ramon Garcia and Roger A. Horn,Matrix mathematics—a second course in linear algebra, second ed., Cambridge Mathematical Textbooks, Cambridge University Press, Cambridge, 2023. MR 4574833

    2023

  13. [13]

    Horn, Florian Luca, and Kuldeep Sarma,Powers of rational matrices, Amer

    Stephan Ramon Garcia, Roger A. Horn, Florian Luca, and Kuldeep Sarma,Powers of rational matrices, Amer. Math. Monthly129(2022), no. 2, 177. MR 4385692

    2022

  14. [14]

    MR 3783061

    Andrew Holbrook,Differentiating the pseudo determinant, Linear Algebra Appl.548(2018), 293–304. MR 3783061

    2018

  15. [15]

    MR 3247242

    Oliver Knill,Cauchy-Binet for pseudo-determinants, Linear Algebra Appl.459(2014), 522–547. MR 3247242

    2014

  16. [16]

    J. C. Rosales and P. A. Garc ´ıa-S´anchez,Numerical semigroups, Developments in Mathematics, vol. 20, Springer, New York, 2009. MR 2549780

    2009

  17. [17]

    Ian Stewart and David Tall,Algebraic number theory and Fermat’s last theorem, 3rd ed., A K Peters / CRC Press, Natick, MA, 2001

    2001

  18. [18]

    J. H. M. Wedderburn,Lectures on matrices, Dover Publications, Inc., New York, 1964. MR 168568 DEPARTMENT OFMATHEMATICS ANDSTATISTICS, POMONACOLLEGE, 610 N. COLLEGEAVE., CLARE- MONT, CA 91711, USA Email address:twcz2023@mymail.pomona.edu Email address:jfwv2022@mymail.pomona.edu Email address:stephan.garcia@pomona.edu URL:https://stephangarcia.sites.pomona....

    1964