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Almost norming vertices for homogeneous polynomials on the cube
Abstract
Let m≥1 be fixed. We consider the space 𝒫m(ℝn) of real m-homogeneous polynomials on ℝn, endowed with the standard Gaussian measure γm,n associated with the Bombieri norm. We study how well the norm of a typical polynomial on the unit ball of ℓ∞n, namely the cube [−1,1]n, can be recovered from its values at the vertices. For P∈𝒫m(ℝn), set M(P)=maxx∈[−1,1]n|P(x)|, V(P)=maxε∈{−1,1}n|P(ε)|. If Pn is chosen according to γm,n, we prove that the relative loss 1−V(Pn)/M(Pn) is of order at most n−1/2 in probability. Consequently, for every 0<β<1/2, γm,n{ P∈𝒫m(ℝn): V(P)≥ (1−n−β)M(P) } ⟶1 as n→∞. Thus, with respect to the Bombieri Gaussian measure, the vertices of the cube are asymptotically norming for homogeneous polynomials of fixed degree.
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BibTeX
@misc{pinasco2026almost,
title = {{Almost norming vertices for homogeneous polynomials on the cube}},
author = {Damián Pinasco and Ignacio Zalduendo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32526},
eprint = {2609.32526},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32526},
}
RIS
TY - UNPB
TI - Almost norming vertices for homogeneous polynomials on the cube
AU - Damián Pinasco
AU - Ignacio Zalduendo
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32526
UR - https://arxiv.org/abs/2609.32526
AB - Let m≥1 be fixed. We consider the space 𝒫_m(ℝ^n) of real m-homogeneous polynomials on ℝ^n, endowed with the standard Gaussian measure γ_m,n associated with the Bombieri norm. We study how well the norm of a typical polynomial on the unit ball of ℓ_∞^n, namely the cube [−1,1]^n, can be recovered from its values at the vertices. For P∈𝒫_m(ℝ^n), set M(P)=max_x∈[−1,1]^n|P(x)|, V(P)=max_ε∈{−1,1}^n|P(ε)|. If P_n is chosen according to γ_m,n, we prove that the relative loss 1−V(P_n)/M(P_n) is of order at most n^−1/2 in probability. Consequently, for every 0<β<1/2, γ_m,n{ P∈𝒫_m(ℝ^n): V(P)≥ (1−n^−β)M(P) } ⟶1 as n→∞. Thus, with respect to the Bombieri Gaussian measure, the vertices of the cube are asymptotically norming for homogeneous polynomials of fixed degree.
ER -