Strong Persistence Index and Fluctuations in Colon Powers of Monomial Ideals

Nasernejad, Mehrdad; Toledo, Jonathan T.

doi:10.3390/math14101705

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Strong Persistence Index and Fluctuations in Colon Powers of Monomial Ideals

by

Mehrdad Nasernejad

^1,2,*

and

Jonathan T. Toledo

³

¹

Univ. Artois, UR 2462, Laboratoire de Mathématique de Lens (LML), F-62300 Lens, France

²

Université Caen Normandie, ENSICAEN, CNRS, Normandie Université, GREYC UMR 6072, F-14000 Caen, France

³

INFOTEC Centro de Investigación e Innovación en Información y Comunicación, Ciudad de México 14050, Mexico

^*

Author to whom correspondence should be addressed.

Mathematics 2026, 14(10), 1705; https://doi.org/10.3390/math14101705

Submission received: 13 April 2026 / Revised: 11 May 2026 / Accepted: 13 May 2026 / Published: 15 May 2026

(This article belongs to the Special Issue Advanced Methods of Polynomial Ideal Computation)

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Abstract

Let I be an ideal in a commutative Noetherian ring R. We say that a positive integer

ℓ_{0}

is the strong persistence index of I if

ℓ_{0}

is the smallest integer such that

(I^{ℓ + 1} :_{R} I) = I^{ℓ}

for all

ℓ \geq ℓ_{0}

. The first aim of this paper is to study this notion for monomial ideals. We also introduce the notion of fluctuation in colon powers if there exist positive integers

a < b < c

such that at least one of the following cases occurs: (i)

(I^{a} : I) = I^{a - 1}, (I^{b} : I) \neq I^{b - 1}, but (I^{c} : I) = I^{c - 1} .

(ii)

(I^{a} : I) \neq I^{a - 1}, (I^{b} : I) = I^{b - 1}, but (I^{c} : I) \neq I^{c - 1} .

The second purpose of this work is to study this phenomenon for monomial ideals.

Keywords: monomial ideals; associated primes; strong persistence index; fluctuation in colon powers

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MDPI and ACS Style

Nasernejad, M.; Toledo, J.T. Strong Persistence Index and Fluctuations in Colon Powers of Monomial Ideals. Mathematics 2026, 14, 1705. https://doi.org/10.3390/math14101705

AMA Style

Nasernejad M, Toledo JT. Strong Persistence Index and Fluctuations in Colon Powers of Monomial Ideals. Mathematics. 2026; 14(10):1705. https://doi.org/10.3390/math14101705

Chicago/Turabian Style

Nasernejad, Mehrdad, and Jonathan T. Toledo. 2026. "Strong Persistence Index and Fluctuations in Colon Powers of Monomial Ideals" Mathematics 14, no. 10: 1705. https://doi.org/10.3390/math14101705

APA Style

Nasernejad, M., & Toledo, J. T. (2026). Strong Persistence Index and Fluctuations in Colon Powers of Monomial Ideals. Mathematics, 14(10), 1705. https://doi.org/10.3390/math14101705

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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Strong Persistence Index and Fluctuations in Colon Powers of Monomial Ideals

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