Characteristic drops for high-order vanishing on the hypercube
Abstract
Let F be a field, let 0≤ ℓ≤ k−2, and suppose that n≥ k−1. We determine the minimum degree of a polynomial in F[x1,…,xn] that vanishes to order at least k at every nonzero vertex of the Boolean cube and to order exactly ℓ at the origin. The answer is n+2k−2−ρF(k−ℓ), where ρF(s) is the least number of positive integers summing to s−1 whose corresponding Catalan numbers are nonzero in F. The proof gives an explicit basis of the reduced vanishing space. In this basis the top-degree map is diagonal, with Catalan numbers on the diagonal; a refinement using block coordinates shows that the basis is compatible with polynomial degree. In odd characteristic the degree is either n+2k−3 or n+2k−4, according as Ck−ℓ−2 is nonzero or zero. In characteristic $2$ it is n+2k−2−s2(k−ℓ−1), where s2 denotes binary digit sum. Thus the first characteristic drop, and all later drops, are determined exactly.
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BibTeX
@misc{menezes2026characteristic,
title = {{Characteristic drops for high-order vanishing on the hypercube}},
author = {Dean Menezes},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19009},
eprint = {2609.19009},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19009},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Characteristic drops for high-order vanishing on the hypercube
AU - Dean Menezes
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.19009
UR - https://arxiv.org/abs/2609.19009
AB - Let F be a field, let 0≤ ℓ≤ k−2, and suppose that n≥ k−1. We determine the minimum degree of a polynomial in F[x_1,…,x_n] that vanishes to order at least k at every nonzero vertex of the Boolean cube and to order exactly ℓ at the origin. The answer is n+2k−2−ρ_F(k−ℓ), where ρ_F(s) is the least number of positive integers summing to s−1 whose corresponding Catalan numbers are nonzero in F. The proof gives an explicit basis of the reduced vanishing space. In this basis the top-degree map is diagonal, with Catalan numbers on the diagonal; a refinement using block coordinates shows that the basis is compatible with polynomial degree. In odd characteristic the degree is either n+2k−3 or n+2k−4, according as C_k−ℓ−2 is nonzero or zero. In characteristic $2$ it is n+2k−2−s_2(k−ℓ−1), where s_2 denotes binary digit sum. Thus the first characteristic drop, and all later drops, are determined exactly.
ER -