各向异性Maxwell系统中介电常数与磁导率的通用恢复
Generic Recovery of Permittivity and Permeability in Anisotropic Maxwell Systems
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中文总结 AI 辅助
该研究针对各向异性电磁介质本构张量反演问题，利用几何不变量理论分析菲涅耳曲面与介电、磁导率张量的关系，证明了反问题的通用唯一性，拓展了仿射几何不变量理论的应用场景。
中文摘要 AI 辅助
我们研究从菲涅耳曲面（即控制电磁波传播的Maxwell方程的特征簇）恢复无磁电耦合（非手性）的均匀各向异性电磁介质本构张量的反问题。对于已知的各向同性磁导率（归一化为$μ= I$），我们证明菲涅耳曲面唯一确定介电张量$\varepsilon$，且当且仅当$\varepsilon$有重特征值时，相关的菲涅耳多项式可约。对于一般的正定对称张量$(\varepsilon,μ)$，我们证明菲涅耳多项式在$\mathbb{C}$上通常是不可约的，并确定了使其保持不变的自然规范对称性。利用几何不变量理论——现代代数几何的一个有力工具，我们构造了参数空间在该规范作用下的仿射商，并证明诱导的菲涅耳多项式映射到其像上是双有理的。我们推断，在一个恰当的实代数例外集之外，实菲涅耳曲面在规范等价意义下确定$(\varepsilon,μ)$。这确立了该反问题的通用唯一性，且据我们所知，为仿射几何不变量理论在PDE反问题的规范自由度方面提供了新的应用。
英文摘要
We study the inverse problem of recovering the constitutive tensors of a homogeneous anisotropic electromagnetic medium without magnetoelectric coupling (non-chiral) from its Fresnel surface, the characteristic variety of Maxwell's equations governing electromagnetic wave propagation. For known isotropic permeability, normalized to $μ= I$, we prove that the Fresnel surface uniquely determines the permittivity tensor $\varepsilon$, and that the associated Fresnel polynomial is reducible precisely when $\varepsilon$ has a repeated eigenvalue. For general, positive-definite symmetric tensors $(\varepsilon,μ)$, we prove that the Fresnel polynomial is generically irreducible over $\mathbb{C}$ and we identify the natural gauge symmetry under which it is invariant. Using geometric invariant theory, a powerful tool of modern algebraic geometry, we construct an affine quotient of the parameter space by this gauge action and prove that the induced Fresnel-polynomial map is birational onto its image. We deduce that, outside a proper real algebraic exceptional set, the real Fresnel surface determines $(\varepsilon,μ)$ up to gauge. This establishes generic uniqueness for the inverse problem and, to our knowledge, provides a new application of affine geometric invariant theory to gauge freedom in a PDE inverse problem.
