Local Structure of Boundary-Normal Resultants for Hyperbolic Characteristic Polynomials

抄録

We study the local structure of the boundary-normal resultant
\[R_{\partial}(\xi_1,\xi')=\Res_{\xi_n}(P,P_{\xi_n})\]
associated with the boundary $x_n=0$, where $\xi_n$ is the Fourier
frequency associated with the boundary-normal coordinate $x_n$.
Here and throughout, ``normal'' refers exclusively to the direction
perpendicular to the boundary $\{x_n=0\}$. For a real-analytic
characteristic germ $P$ at a multiple characteristic point,
let $Q=\inop P$ denote its tangent polynomial.
Under the boundary-normal noncharacteristic condition $Q(e_n)\neq0$,
we establish three main results and give a contrasting example.
First, after localization at a fixed multiple boundary-normal root,
cross-resultants between distinct root clusters are local units, and the
initial form of the resultant factors as a product of the tangent
discriminants of the individual clusters. Second, for a geometrically
regular, nonglancing, first-order normally separated cluster of arbitrary
multiplicity $m$, the tangent discriminant is a product of squares of real
linear forms and is hyperbolic with respect to suitable directions. In this
setting, the real boundary-normal branch-collision locus is a finite union
of analytic hypersurfaces, and its Bouligand tangent cone coincides with
the algebraic tangent cone $\{\Disc_{\xi_n}Q=0\}$. Third, for a genuinely
linearly splitting double characteristic in the standard symmetrizable
setting, we give an explicit formula for the reduced boundary-normal
discriminant on the effective transverse quotient. The resulting E/H
dichotomy follows from classical Lorentzian signature theory; in the
standard setting, the reduced quotient has dimension two, so the signatures
are $(2,0)$ and $(1,1)$. Finally, an explicit family shows that a product of
two characteristic factors can have an ultrahyperbolic factor of signature
$(p,q)$ with $p,q\ge2$ in its boundary-normal resultant, even under strict
hyperbolicity in the time-frequency variable, because the two factors share
the same boundary-normal root at the base point and their cross-resultant is
not a local unit. The genuinely linearly splitting case of multiplicity
$m\ge3$ remains open.