Discrete Unique Continuation on Simplex

For integers $N\ge 0$ and $n\ge 2$, let $${\mathrm{\Delta }}_N^{(n)}=\{\alpha \in {\mathbb{Z}}_{\ge 0}^n:{\alpha }_1+\cdots +{\alpha }_n=N\}\mathrm{.}$$ We study discrete unique continuation for functions on this lattice simplex. For an integer $R\ge 1$, let $g:{\mathrm{\Delta }}_{nR}^{(n)}\to \mathbb{R}$ satisfy $$\sum _{i=1}^{n}g(\beta +e_i)=0,\beta \in {\mathrm{\Delta }}_{nR-1}^{(n)},$$ where $e_i$ is the $i$th standard basis vector. We prove that nonvanishing at the center implies $$\mathrm{\mid }\mathrm{supp}g\mathrm{\mid }\ge c_n{R}^{\lceil n\mathrm{/}2\rceil }\mathrm{.}$$ Here $\mathrm{supp}g$ is the set of points where $g$ is nonzero, and $c_n>0$ depends only on $n$. The exponent $\lceil n\mathrm{/}2\rceil$ 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.