Adjoints and Canonical Forms of Polytopes

Projectivizations of pointed polyhedral cones \(C\) are positive geometries with canonical forms that look like \(\Omega_C(x) = \frac{A(x)}{B(x)}dx\) with \(A, B\) polynomials. We are going to see that the numerator \(A(x)\) is given by the adjoint polynomial of the dual cone \(C^\vee\), which cancels unwanted poles outside the polytope.

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Date published: Wednesday, October 11, 2023