Discrete Unique Continuation on Simplex

Abstract

For integers $N\ge 0$ and $n\ge 2$, let \[ Δ_N^{(n)} =\left\{α\in\mathbb Z_{\ge 0}^n: α_1+\cdots+α_n=N\right\}. \] We study discrete unique continuation for functions on this lattice simplex. For an integer $R\ge 1$, let $g:{\Delta }_{nR}^{(n)}\to \mathbb{R}$ satisfy \[ \sum_{i=1}^n g(β+e_i)=0, \qquad β\inΔ_{nR-1}^{(n)}, \] where $e_i$ is the $i$th standard basis vector. We prove that nonvanishing at the center implies \[ |\operatorname{supp} g|\ge c_n R^{\lceil n/2\rceil}. \] Here $\mathrm{supp}g$ is the set of points where $g$ is nonzero, and $c_n>0$ depends only on $n$. The exponent $\lceil n/2\rceil$ 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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