Monodromy of the complexified Legendre transform

Compute the discriminant variety and monodromy representation of the complexified Legendre transform for the explicit definite- and Lorentzian-signature examples constructed in the paper, and determine their relation to the local exponents in Proposition 11.37.

Background

For real-analytic difference-of-convex potentials, the Legendre map complexifies to a multi-valued holomorphic inverse away from the image of the discriminant variety. The paper establishes the existence of this structure but does not calculate its global monodromy.

The unresolved problem concerns the explicit examples in Sections 11.5–11.6 and whether their monodromy can be related directly to the finite-order vanishing exponents governing Łojasiewicz-controlled convergence near critical hypersurfaces.