Ring of volume invariants from iterated integral signatures

Determine the ring of volume invariants derived from signatures of piecewise-linear paths on cyclic polytopes and prove Lotter–Preiß (Conjecture 4.1).

Background

Iterated integral signatures of paths encode geometric information in tensorial form. For cyclic polytopes—central to amplituhedra—these signatures can express volume-related invariants. Characterizing the ring of such invariants would systematize which linear combinations of signature entries capture intrinsic polytope geometry, linking stochastic analysis with combinatorial geometry.