Inertia and Elementary Theories of Hermitian Nilpotent Operators
Source: hexagonmath.org/abs/2611.00089v1
Let \(V\) be a finite-dimensional Hermitian space over an algebraically closed field with a nonidentity involution, and suppose its form \(h\) is nondegenerate. For a self-adjoint nilpotent operator \(N\), we recover the multiplicity of each signed Jordan block from finite differences of the ranks and signature differences of the forms \(h(x,N^j y)\). On a fixed Hermitian space these inertias classify unitary conjugacy. Across scalar fields they classify elementary equivalence in the four-sorted language of scalars, vectors, endomorphisms, and projective points with \(N\) named. At fixed…
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