Structure of Polyzetas and Explicit Representation on Transcendence Bases of Shuffle and Stuffle Algebras

TL;DR

This paper investigates the algebraic structure of polyzetas, constructing dual bases and algorithms to compute relations and explicit representations within shuffle and quasi-shuffle algebra frameworks.

Contribution

It introduces dual bases for polyzetas and new algorithms for identifying polynomial relations and explicit representations in their algebraic setting.

Findings

Constructed dual bases for polyzetas in shuffle and quasi-shuffle algebras

Developed algorithms to compute polynomial relations among polyzetas

Provided explicit data structures for representing polyzetas in terms of irreducible elements

Abstract

Polyzetas, indexed by words, satisfy shuffle and quasi-shuffle identities. In this respect, one can explore the multiplicative and algorithmic (locally finite) properties of their generating series. In this paper, we construct pairs of bases in duality on which polyzetas are established in order to compute local coordinates in the infinite dimensional Lie groups where their non-commutative generating series live. We also propose new algorithms leading to the ideal of polynomial relations, homogeneous in weight, among polyzetas (the graded kernel) and their explicit representation (as data structures) in terms of irreducible elements.