标题： 周期超曲面与李杨多项式 显示英文标题

标题： Periodic Hypersurfaces and Lee-Yang Polynomials

摘要： 我们证明了一个傅里叶准则，即一个$\mathbb{Z}^n$周期的$C^{1+\epsilon}$级曲面$\Sigma\subset\mathbb{R}^n$在意义上是代数的，即$\Sigma=\Sigma(p):=\{x\in\mathbb{R}^n : p(e^{2\pi i x_1},\dots,e^{2\pi i x_n})=0\}$对某个多项式$p$。 更准确地说，每个方向$\ell\in\mathbb{R}^n$在$\Sigma$上通过$dm_{\ell}(x)=|\langle\ell,\hat n(x)\rangle|\,d\sigma(x)$定义了一个周期性测度$m_{\ell}$，其中$\hat n(x)$是$\Sigma$的单位法向量，$\sigma$是其$n-1$维的曲面测度。 我们的傅里叶准则对于$\Sigma$是存在一个具有Q-线性无关条目的$\ell\in\mathbb{R}^n$和一个闭锥$C\subset\mathbb{R}^n$，使得对于所有非零$v\in C$，有$\langle\ell,v\rangle>0$，并且对于$k\in\mathbb{Z}^n$，我们有$\widehat{m}_{\ell}(k)=\int_{\Sigma/\mathbb{Z}^n}e^{2\pi i\langle k,x\rangle}\,dm_{\ell}(x)=0$对于每个$k\notin C\cup(-C)$。 使用Meyers的术语，这表示$(m_{\ell},C)$是一个“灯塔”。 在该假设下可以证明$\Sigma=\Sigma(p)$，并且经过适当的单项变换后$p$可以作为Lee-Yang多项式。 该证明依赖于对所有一维傅里叶准晶体的最新表征。 显示英文摘要

摘要： We prove a Fourier criterion for a $\mathbb{Z}^n$-periodic $C^{1+\epsilon}$-hypersurface $\Sigma\subset\mathbb{R}^n$ to be algebraic in the sense that $\Sigma=\Sigma(p):=\{x\in\mathbb{R}^n : p(e^{2\pi i x_1},\dots,e^{2\pi i x_n})=0\}$ for some polynomial $p$. More precisely, each direction $\ell\in\mathbb{R}^n$ defines a periodic measure $m_{\ell}$ on $\Sigma$ by $dm_{\ell}(x)=|\langle\ell,\hat n(x)\rangle|\,d\sigma(x)$, with $\hat n(x)$ the unit normal to $\Sigma$ and $\sigma$ its $n-1$ dimensional surface measure. Our Fourier criterion for $\Sigma$ is that there exists $\ell\in\mathbb{R}^n$ with Q-linearly independent entries and a closed cone $C\subset\mathbb{R}^n$ for which $\langle\ell,v\rangle>0$ for all nonzero $v\in C$, such that for $k\in\mathbb{Z}^n$ we have $\widehat{m}_{\ell}(k)=\int_{\Sigma/\mathbb{Z}^n}e^{2\pi i\langle k,x\rangle}\,dm_{\ell}(x)=0$ for every $k\notin C\cup(-C)$. Using Meyers terminology this says $(m_{\ell},C)$ is a "lighthouse". Under this hypothesis one shows $\Sigma=\Sigma(p)$, and after a suitable monomial change $p$ can be taken as a Lee-Yang polynomial. The proof relies on a recent characterization of all one-dimensional Fourier quasicrystals.