Cyclic polytopes through the lens of iterated integrals

TL;DR

This paper explores how the volume of cyclic polytopes can be expressed through iterated integrals and investigates invariants under automorphism subgroup actions, revealing infinitely many algebraically independent invariants.

Contribution

It introduces new polynomial invariants of cyclic polytopes derived from iterated integrals and proves their algebraic independence within the shuffle algebra.

Findings

Infinite algebraically independent invariants identified

New polynomial attributes of cyclic polytopes discovered

Connections between integrals and polytope symmetries established

Abstract

The volume of a cyclic polytope can be obtained by forming an iterated integral along a suitable piecewise linear path running through its edges. Different choices of such a path are related by the action of a subgroup of the combinatorial automorphisms of the polytope. Motivated by this observation, we look for other linear combinations of iterated integrals that are invariant under the subgroup action. This yields interesting polynomial attributes of the cyclic polytope. We prove that there are infinitely many of these invariants which are algebraically independent in the shuffle algebra.