[Paper Review] Conical zeta values and their double subdivision relations

Li Guo, Sylvie Paycha+1 more|arXiv (Cornell University)|Jan 15, 2013

Advanced Mathematical Identities12 references3 citations

TL;DR

This paper introduces conical zeta values (CZVs) as a geometric generalization of multiple zeta values (MZVs) using convex cones, establishing open and closed cone subdivision relations that generalize the quasi-shuffle and shuffle relations of MZVs. The key contribution is the formulation of the double subdivision relation for CZVs and the extended double subdivision relation conjecture, unifying MZV relations in a geometric framework and showing that CZVs, LZVs, and SZVs span the same rational vector space.

ABSTRACT

We introduce the concept of a conical zeta value as a geometric generalization of a multiple zeta value in the context of convex cones. The quasi-shuffle and shuffle relations of multiple zeta values are generalized to open cone subdivision and closed cone subdivision relations respectively for conical zeta values. In order to achieve the closed cone subdivision relation, we also interpret linear relations among fractions as subdivisions of decorated closed cones. As a generalization of the double shuffle relation of multiple zeta values, we give the double subdivision relation of conical zeta values and formulate the extended double subdivision relation conjecture for conical zeta values.

Motivation & Objective

• To generalize multiple zeta values (MZVs) to conical zeta values (CZVs) via convex cones, extending their algebraic and geometric structure.
• To establish open cone subdivision relations as the geometric analog of the quasi-shuffle product for MZVs.
• To define closed cone subdivision relations using linear relations among fractions, generalizing the shuffle product for MZVs.
• To formulate the double subdivision relation for CZVs, mirroring the double shuffle relation of MZVs.
• To propose the extended double subdivision relation conjecture, providing a unified framework for algebraic relations among CZVs.

Proposed method

• Define conical zeta values (CZVs) as zeta functions over open convex cones generated by integer vectors, with convergence conditions on complex parameters.
• Introduce open cone subdivisions to generalize the quasi-shuffle (stuffle) product, encoding CZV relations via geometric decomposition of cones.
• Use decorated closed cones and pure fractions to model linear relations among rational functions, enabling the derivation of closed cone subdivision relations.
• Establish the closed cone subdivision relation as a geometric realization of the shuffle product, linking algebraic shuffle relations to cone geometry.
• Formulate the double subdivision relation by combining open and closed cone subdivision relations, generalizing the double shuffle relation of MZVs.
• Relate CZVs to Shintani zeta values (SZVs) via matrix representations of cones, showing that CZVs are rational linear combinations of SZVs.

Experimental results

Research questions

• RQ1 How can multiple zeta values be generalized geometrically using convex cones?
• RQ2 What are the analogs of the quasi-shuffle and shuffle relations for conical zeta values?
• RQ3 Can the double shuffle relation of MZVs be extended to a double subdivision relation in the context of conical zeta values?
• RQ4 What is the relationship between conical zeta values, linear zeta values (LZVs), and Shintani zeta values (SZVs) over the rationals?
• RQ5 Is the extended double subdivision relation conjecture valid for all conical zeta values, and how does it unify algebraic and geometric structures?

Key findings

• Conical zeta values (CZVs) generalize multiple zeta values (MZVs) by defining zeta functions over open convex cones in the integer lattice.
• Open cone subdivisions yield a geometric analog of the quasi-shuffle product, generalizing the stuffle relation of MZVs.
• Closed cone subdivisions, derived from linear relations among fractions, generalize the shuffle product of MZVs.
• The double subdivision relation for CZVs unifies the open and closed cone subdivision relations, generalizing the double shuffle relation of MZVs.
• All convergent CZVs, LZVs, and SZVs span the same rational vector space, i.e., $ oxed{{b Q} ext{CZV} = {b Q} ext{LZV} = {b Q} ext{SZV}} $.
• Every convergent CZV is a rational linear combination of Shintani zeta values (SZVs), and every convergent maximal rank SZV is a rational linear combination of LZVs, establishing a complete rational equivalence among these zeta value classes.