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Discrete Unique Continuation on Simplex
Abstract
For integers N≥0 and n≥2, let ΔN(n) ={α∈ℤ≥ 0n: α1+⋯+αn=N}. We study discrete unique continuation for functions on this lattice simplex. For an integer R≥1, let g:ΔnR(n)→ℝ satisfy ∑i=1n g(β+ei)=0, β∈ΔnR−1(n), where ei is the ith standard basis vector. We prove that nonvanishing at the center implies |supp g|≥ cn R⌈ n/2⌉. Here supp g is the set of points where g is nonzero, and cn>0 depends only on n. The exponent ⌈ n/2⌉ is optimal. The proof represents the values of g as polynomial coefficients and uses a Pascal uncertainty principle, which bounds from below the total number of nonzero coefficients of a one-variable polynomial and its unit translate in terms of their degree.
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BibTeX
@misc{li2026discrete,
title = {{Discrete Unique Continuation on Simplex}},
author = {Linjun Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.02707},
eprint = {2608.02707},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.02707},
}
RIS
TY - UNPB
TI - Discrete Unique Continuation on Simplex
AU - Linjun Li
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2608.02707
UR - https://arxiv.org/abs/2608.02707
AB - For integers N≥0 and n≥2, let Δ_N^(n) ={α∈ℤ_≥ 0^n: α_1+⋯+α_n=N}. We study discrete unique continuation for functions on this lattice simplex. For an integer R≥1, let g:Δ_nR^(n)→ℝ satisfy ∑_i=1^n g(β+e_i)=0, β∈Δ_nR−1^(n), where e_i is the ith standard basis vector. We prove that nonvanishing at the center implies |supp g|≥ c_n R^⌈ n/2⌉. Here supp g is the set of points where g is nonzero, and c_n>0 depends only on n. The exponent ⌈ n/2⌉ is optimal. The proof represents the values of g as polynomial coefficients and uses a Pascal uncertainty principle, which bounds from below the total number of nonzero coefficients of a one-variable polynomial and its unit translate in terms of their degree.
ER -