Ansgar
Freyer
FU Berlin
The adjoint degree and translation scissors congruence
Abstract.
The adjoint polynomial of a convex polytope P entails many nice information on the geometry and combinatorics of P. It gives strong insights in the structure of the residual arrangement of P, that is, the hyperplane arrangement obtained by extending the facets. The adjoint appears, moreover, if one seeks to find "nice" convex combinations for interior points of P, or as the numerator of the canonical form of P.
In this talk, we are concerned with the degree of the adjoint polynomial. In the projective setting, this degree is always m-d-1, where m is the number of facets of P. For non-generic polytopes in an affine space, the degree may be lower, which could be seen as a hint on parallelisms among the faces of P.
Our main tool to study the degree drop of the adjoint is a modification of the so-called canonical form. That way we obtain a translation invariant valuation that vanishes exactly when the adjoint degree is "lower than usual". We will see that this function is, in addition, 1-homogeneous and therefore commutes with Minkowski addition. From these properties we derive structural results on polytopes for which the adjoint degree is low, including a full characterization of all possible degrees in dimension up to 3.
This is a joint work with Tom Baumbach, Julian Weigert and Martin Winter.