Generic Recovery of Permittivity and Permeability in Anisotropic Maxwell Systems

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 $\mu = I$, 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 $(\varepsilon,\mu)$, 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 $(\varepsilon,\mu)$ 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.