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Two Infinite Families of Regular Sequences of Power Sums in Three Variables
Abstract
Let S=ℂ[x,y,z], pm=xm+ym+zm. For every r≥1, we prove that pr, pr+1, p3r+1 and pr, pr+1, p3r+2 are regular sequences exactly when rnot≡1pmod3. This proves the Conca--Krattenthaler--Watanabe prediction for two infinite families. The proof is elementary. Four simple reductions modulo (pr,pr+1) reduce the problem to the two equations x+y+z=0 and xy+xz+yz=0.
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BibTeX
@misc{abedelfatah2026infinite,
title = {{Two Infinite Families of Regular Sequences of Power Sums in Three Variables}},
author = {Abed Abedelfatah},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07932},
eprint = {2609.07932},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07932},
}
RIS
TY - UNPB
TI - Two Infinite Families of Regular Sequences of Power Sums in Three Variables
AU - Abed Abedelfatah
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07932
UR - https://arxiv.org/abs/2609.07932
AB - Let S=ℂ[x,y,z], p_m=x^m+y^m+z^m. For every r≥1, we prove that p_r, p_r+1, p_3r+1 and p_r, p_r+1, p_3r+2 are regular sequences exactly when rnot≡1pmod3. This proves the Conca--Krattenthaler--Watanabe prediction for two infinite families. The proof is elementary. Four simple reductions modulo (p_r,p_r+1) reduce the problem to the two equations x+y+z=0 and xy+xz+yz=0.
ER -