Parity-Sensitive Fourier Uncertainty and Zero-Set Rigidity on the Finite Parabola

Abstract

Let $p\ge5$ be prime, let $R\subset\mathbb F_p$ satisfy $1\le |R|\le p-1$, and set \[ F(a,b)=\sum_{x\in R}c_xω^{ax^2+bx}, \qquad (a,b)\in\mathbb F_p^2, \qquad c_x\ne0, \] where $ω=e^{2πi/p}$. We prove a parity-sensitive uncertainty principle for complex Fourier spectra supported on the finite parabola. If $|R|=2r$, then \[ |Z(F)|\le p+2r-2, \] and the bound is sharp for every even support size $2\le |R|\le p-1$, including the endpoint $|R|=p-1$. If $|R|=2r+1$, then \[ |Z(F)|\le \min\{p+2r-2,\,r(r+1)\}. \] Thus fixed odd spectral sparsity forces a number of zeros bounded independently of $p$, whereas sufficiently large even zero sets are rigid: for even support $|R|=2r$, if $|Z(F)|>r(r+1)$, then $F$ vanishes identically on a nonvertical affine line and has at most $r(r-1)$ further zeros. The proof reduces arbitrary complex coefficients to cyclotomic data and uses a truncated $(1-ω)$-adic expansion. The first nonzero finite-field jets satisfy \[ \partial_b^2Q_j=\partial_aQ_{j-1}, \] turning the two-dimensional zero problem into a multiplicity and component-persistence problem for algebraic curves. As an application, we solve the total-support uncertainty problem for the standard complete set of $p+1$ mutually unbiased bases in $\mathbb C^p$: \[ \min_{0\neψ\in\mathbb C^p} \sum_j |\operatorname{supp}_{\mathcal B_j}(ψ)| =p^2-p+2. \] We also classify all extremizing projective states and give an exact enumeration formula.