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On the Algebraic Bases of Polyzetas

Vincel Hoang Ngoc Minh

TL;DR

The paper builds a rigorous algebraic-combinatorial framework for polyzetas by embedding multiple zeta values, polylogarithms, and harmonic sums into shuffle and quasi-shuffle (stuffle) algebras and leveraging PBW/CQMM theory. It introduces confluent rewriting systems that encode the kernels of the zeta polymorphism and shows that the algebra of polyzetas is graded and freely generated by irreducible elements, with these irreducibles being ${\mathbb Q}$-algebraically independent and yielding transcendence results for polyzetas (and independence for $\pi^2$ from odd zeta values). An Abel-type renormalization links noncommutative generating series of polylogarithms and harmonic sums, enabling a transfer of analytic behavior to algebraic structure and establishing a bijection between irreducible generators and zeta-values via the LocalCoordinateIdentification algorithm. The work provides explicit irreducible sets up to weight 12 and supports conjectures on the graded decomposition of polyzetas, offering a concrete computational route to determine the algebraic relations among polyzetas and their generators. Overall, the approach unifies noncommutative generating series, rewriting systems, and number-theoretic independence results to advance understanding of the algebraic bases of polyzetas and their applications in algebraic geometry, knot invariants, and mathematical physics.

Abstract

Two confluent rewriting systems in noncommutatives polynomials are constructed using the equations allowing the identification of the local coordinates (of second kind) of the graphs of the $ζ$ polymorphism as being (shuffle or quasi-shuffle) characters and bridging two algebraic structures of polyzetas. In each system, the left side of each rewriting rule corresponds to the leading monomial of the associated homogeneous in weight polynomial while the right side is canonically represented on the Q-algebra generated by irreducible terms which encode an algebraic basis of the Q-algebra of polyzetas. These polynomials are totally lexicographically ordered and generate the kernels of the $ζ$ polymorphism meaning that the Q-free algebra of polyzetas is graded and the irreducible polyzetas are transcendent numbers, Q-algebraically independent, and then $π$ 2 is Q-algebraically independent on odd zeta values (so does $π$).

On the Algebraic Bases of Polyzetas

TL;DR

The paper builds a rigorous algebraic-combinatorial framework for polyzetas by embedding multiple zeta values, polylogarithms, and harmonic sums into shuffle and quasi-shuffle (stuffle) algebras and leveraging PBW/CQMM theory. It introduces confluent rewriting systems that encode the kernels of the zeta polymorphism and shows that the algebra of polyzetas is graded and freely generated by irreducible elements, with these irreducibles being -algebraically independent and yielding transcendence results for polyzetas (and independence for from odd zeta values). An Abel-type renormalization links noncommutative generating series of polylogarithms and harmonic sums, enabling a transfer of analytic behavior to algebraic structure and establishing a bijection between irreducible generators and zeta-values via the LocalCoordinateIdentification algorithm. The work provides explicit irreducible sets up to weight 12 and supports conjectures on the graded decomposition of polyzetas, offering a concrete computational route to determine the algebraic relations among polyzetas and their generators. Overall, the approach unifies noncommutative generating series, rewriting systems, and number-theoretic independence results to advance understanding of the algebraic bases of polyzetas and their applications in algebraic geometry, knot invariants, and mathematical physics.

Abstract

Two confluent rewriting systems in noncommutatives polynomials are constructed using the equations allowing the identification of the local coordinates (of second kind) of the graphs of the polymorphism as being (shuffle or quasi-shuffle) characters and bridging two algebraic structures of polyzetas. In each system, the left side of each rewriting rule corresponds to the leading monomial of the associated homogeneous in weight polynomial while the right side is canonically represented on the Q-algebra generated by irreducible terms which encode an algebraic basis of the Q-algebra of polyzetas. These polynomials are totally lexicographically ordered and generate the kernels of the polymorphism meaning that the Q-free algebra of polyzetas is graded and the irreducible polyzetas are transcendent numbers, Q-algebraically independent, and then 2 is Q-algebraically independent on odd zeta values (so does ).
Paper Structure (5 sections, 9 theorems, 64 equations)

This paper contains 5 sections, 9 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Algebraic combinatorial framework
  3. Polylogarithms, harmonic sums and polyzetas viewed as functions on free monoids and their graphs
  4. From algebraic relations among polyzetas to rewriting systems
  5. Conclusion

Key Result

theorem 1

Let $\varphi_{\pi_1}:(A\langle Y\rangle,\mathop{\tt conc},1_{Y^*})\longrightarrow(A\langle Y\rangle,\mathop{\tt conc},1_{Y^*})$ maps $y_k$ to $\pi_1(y_k)$. It is an automorphism of $A\langle Y\rangle$ realizing an isomorphism of bialgebras between ${\mathcal{H}}_{\mathop{_{^{\sqcup\!\sqcup}}}}(Y)$ a and $\{\Pi_w\}_{w\in Y^*}$ (resp. $\{\Sigma_w\}_{w\in Y^*}$) is image of $\{P_w\}_{w\in Y^*}$ (resp

Theorems & Definitions (16)

  • theorem 1: CM
  • remark 1
  • theorem 2: CM
  • corollary 1: CM
  • theorem 3: Abel like theorem, DaresburyJSC
  • corollary 2: actaVJM
  • remark 2
  • remark 3
  • proposition 1
  • proof
  • ...and 6 more