发表机构
School of Mathematics, Hefei University of Technology（合肥工业大学数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对有限抛物线上的复傅里叶谱，证明了奇偶敏感的零点个数上界，并揭示了大偶数零点集的刚性结构，进而解决了互不偏基的总支撑不确定性极值问题。
AI 中文摘要
设 $p\ge5$ 为素数，$R\subset\mathbb F_p$ 满足 $1\le |R|\le p-1$，并设 \\[ F(a,b)=\sum_{x\in R}c_x\omega^{ax^2+bx}, \qquad (a,b)\in\mathbb F_p^2, \qquad c_x\ne0, \\] 其中 $\omega=e^{2\pi i/p}$。我们证明了支撑在有限抛物线上的复傅里叶谱的一个奇偶敏感不确定性原理。若 $|R|=2r$，则 \\[ |Z(F)|\le p+2r-2, \\] 且该界对每个偶数支撑大小 $2\le |R|\le p-1$（包括端点 $|R|=p-1$）都是尖锐的。若 $|R|=2r+1$，则 \\[ |Z(F)|\le \min\{p+2r-2,\\,r(r+1)\}. \\] 因此，固定的奇数谱稀疏度迫使零点数目有一个与 $p$ 无关的界，而足够大的偶数零点集是刚性的：对于偶数支撑 $|R|=2r$，若 $|Z(F)|>r(r+1)$，则 $F$ 在一条非垂直仿射直线上恒为零，并且至多有 $r(r-1)$ 个额外零点。证明将任意复系数约化为分圆数据，并使用截断的 $(1-\omega)$-进展开。第一个非零有限域 jets 满足 \\[ \partial_b^2Q_j=\partial_aQ_{j-1}, \\] 从而将二维零点问题转化为代数曲线的重数和分量持续性问题。作为应用，我们解决了 $\mathbb C^p$ 中标准完备的 $p+1$ 个互不偏基的总支撑不确定性问题：\\[ \min_{0\ne\psi\in\mathbb C^p} \sum_j |\operatorname{supp}_{\mathcal B_j}(\psi)| =p^2-p+2. \\] 我们还对所有极值投影态进行了分类，并给出了精确的枚举公式。
英文摘要
We determine the sharp total-support uncertainty for the standard complete set of mutually unbiased bases in every prime dimension at least five, proving that the minimum is \(T_s(p)=p^2-p+2\). This resolves all prime dimensions left open by Fiorentino and Weigert, beginning with dimension eleven, and gives a uniform analytic determination of the optimal lower bound in dimension seven, where the earlier determination was partly numerical. The main new ingredient is a sharp zero-count theorem for arbitrary complex Fourier data supported on the finite parabola. A cyclotomic reduction and truncated local expansion produce a first nonzero finite-field jet that is affine in the transverse variable, reducing the planar zero count to a one-dimensional multiplicity problem. We then classify all extremizing projective states and derive an exact enumeration formula. In dimension seven, this gives analytic four-point extremizers and exactly \(12348\) such rays, completing the analytic description and count left open in the earlier work.
