The Collatz map analogue in polynomial rings and in completions

Angelot Behajaina, Elad Paran

نتاج البحث: نشر في مجلةمقالةمراجعة النظراء

ملخص

We study an analogue of the Collatz map in the polynomial ring R[x], where R is an arbitrary commutative ring. We prove that if R is of positive characteristic, then every polynomial in R[x] is eventually periodic with respect to this map. This extends previous works of the authors and of Hicks, Mullen, Yucas and Zavislak, who studied the Collatz map on Fp[x] and F2[x], respectively. We also consider the Collatz map on the ring of formal power series R[[x]] when R is finite: we characterize the eventually periodic series in this ring, and give formulas for the number of cycles induced by the Collatz map, of any given length. We provide similar formulas for the original Collatz map defined on the ring Z2 of 2-adic integers, extending previous results of Lagarias.

اللغة الأصليةالإنجليزيّة
رقم المقال114273
دوريةDiscrete Mathematics
مستوى الصوت348
رقم الإصدار1
المعرِّفات الرقمية للأشياء
حالة النشرنُشِر - يناير 2025

ملاحظة ببليوغرافية

Publisher Copyright:
© 2024 Elsevier B.V.

بصمة

أدرس بدقة موضوعات البحث “The Collatz map analogue in polynomial rings and in completions'. فهما يشكلان معًا بصمة فريدة.

قم بذكر هذا