Adjoint degrees and scissor congruence for polytopes

Hilbert's Third Problem asks whether for any two (3-dimensional) polytopes there is a way to cut the first one into finitely many pieces and rearrange them to obtain the second one (that is, we ask whether the polytopes are "scissors congruent"). Its resolution by Max Dehn (with a negative answer) marks the beginning of valuation theory; and still today, valuations often provide one of the more elegant approaches to problems in, for example, Ehrhart theory or the geometric theory of polytopes.

In this talk we take a look at everybody's new favorite valuation - the canonical form - and we shall explore what it can teach us about scissors congruence for polytopes. It will turn out that the degree of the adjoint polynomial is a fundamental parameter in this context. We investigate the polytope classes defined by their adjoint degrees.

Joined work with Tom Baumbach, Ansgar Freyer and Julian Weigert