On Least Common Multiples of Polynomials in Z/nZ[x]

Abstract

Let 𝒫(n, D) be the set of all monic polynomials in ℤ/nℤ[x] of degree D. A least common multiple for 𝒫(n, D) is a monic polynomial L ∈ ℤ/nℤ[x] of minimal degree such that f divides L for all f ∈ 𝒫(n, D). A least common multiple for 𝒫(n, D) always exists, but need not be unique; however, its degree is always unique. In this article, we establish some bounds for the degree of a least common multiple for 𝒫(n, D), present constructions for common multiples in ℤ/nℤ[x], and describe a connection to rings of integer-valued polynomials over matrix rings.

Key Words:

• Finite ring
• Least common multiple
• Matrix
• Polynomial

2010 Mathematics Subject Classification:

• Primary 13M10
• Secondary 11C99

ACKNOWLEDGMENT

The author would like to thank the referee for the many helpful suggestions made during the review process.

Notes

Communicated by I. Swanson.