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On an ideal membership problem
Abstract
Suppose f1, …, fn+1 are elements of a regular ring R, where n := dim R. The first author proved that f1n ⋯ fn+1n ∈ (f1n+1, …, fn+1n+1)R; this uses the Brian\c con-Skoda theorem --- no simpler proof is known, as far as we are aware. We show here that this result is optimal in many respects. On the other hand, when the fi are homogeneous general polynomials of a fixed degree in R := 𝔽[x1,…,xn], for 𝔽 a field of characteristic zero, we prove that f1 ⋯ fn+1 ∈ (f12, …, fn+12)R.
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BibTeX
@misc{hochster2026ideal,
title = {{On an ideal membership problem}},
author = {Melvin Hochster and Jack Jeffries and Nawaj KC and Anurag K. Singh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10200},
eprint = {2610.10200},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10200},
}
RIS
TY - UNPB
TI - On an ideal membership problem
AU - Melvin Hochster
AU - Jack Jeffries
AU - Nawaj KC
AU - Anurag K. Singh
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10200
UR - https://arxiv.org/abs/2610.10200
AB - Suppose f_1, …, f_n+1 are elements of a regular ring R, where n := dim R. The first author proved that f_1^n ⋯ f_n+1^n ∈ (f_1^n+1, …, f_n+1^n+1)R; this uses the Brian\c con-Skoda theorem --- no simpler proof is known, as far as we are aware. We show here that this result is optimal in many respects. On the other hand, when the f_i are homogeneous general polynomials of a fixed degree in R := 𝔽[x_1,…,x_n], for 𝔽 a field of characteristic zero, we prove that f_1 ⋯ f_n+1 ∈ (f_1^2, …, f_n+1^2)R.
ER -