[Paper Review] About a conjectured basis for Multiple Zeta Values

J. Kuipers, J.A.M. Vermaseren|arXiv (Cornell University)|May 10, 2011

Advanced Mathematical Identities7 references3 citations

TL;DR

This paper confirms a conjecture that twofold extended basis elements are required for multiple zeta values (MZVs) at weights 27 and 28, using optimized computer algebra techniques in TFORM/FORM. The authors verify that the predicted basis structures—derived from Lyndon words of odd integers greater than one—match explicit computations, confirming that double pushdowns occur at these weights, resolving a long-standing prediction from Broadhurst and Kreimer's generating functionals.

ABSTRACT

We confirm a conjecture about the construction of basis elements for the multiple zeta values (MZVs) at weight 27 and weight 28. Both show as expected one element that is twofold extended. This is done with some lengthy computer algebra calculations using TFORM to determine explicit bases for the MZVs at these weights.

Motivation & Objective

• To verify the existence of twofold extended basis elements for multiple zeta values (MZVs) at weights 27 and 28, as predicted by a revised generating functional from Broadhurst and Kreimer.
• To test whether the conjectured pushdown phenomena—where MZVs can be expressed with reduced depth—coincide with n-fold extended basis constructions.
• To develop and apply computational optimizations in TFORM/FORM to handle the high-complexity calculations required for MZV basis determination at these weights.
• To confirm that the basis elements derived from Lyndon words of odd integers greater than one, with appropriate extensions, fully describe the MZV space at weights 27 and 28.

Proposed method

• The authors use TFORM/FORM computer algebra systems to compute explicit MZV bases at weights 27 and 28, applying stuffle and shuffle relations to reduce MZVs to canonical forms.
• They optimize the computation by storing intermediate results in tables after stuffle reduction, reducing the number of substitutions by a factor of depth D.
• A key innovation is modifying TFORM to distribute entire brackets (not just terms) across workers, enabling cancellations within each worker and drastically reducing disk usage and sorting overhead.
• The method involves constructing basis elements from Lyndon words of odd integers >1 summing to the weight, with n-fold extensions defined by decrementing the first 2n indices and appending 2n ones.
• The resulting bases are checked for minimality in depth and compared to the predictions of table 18 in ref. [3], which is based on the Broadhurst-Kreimer conjecture.
• The authors validate that the basis elements with trailing ones, when transformed via the extension rule, recover the full set of Lyndon words $L_W$ for weights 27 and 28.

Experimental results

Research questions

• RQ1 Does the predicted twofold extended basis structure for MZVs at weight 27 and 28, as per the revised Broadhurst-Kreimer generating functional, actually occur in explicit computations?
• RQ2 Are the pushdown phenomena—where MZVs can be expressed with lower depth—equivalent to the n-fold extended basis constructions predicted by the conjecture?
• RQ3 Can the computational complexity of MZV basis determination at weights 27 and 28 be overcome through algorithmic and system-level optimizations in TFORM/FORM?
• RQ4 To what extent do the basis elements derived from Lyndon words of odd integers >1, with controlled extensions, reproduce the full MZV space at these weights?

Key findings

• The basis for MZVs at weight 27 contains 149 elements, and at weight 28 contains 160 elements, both matching the predictions of table 18 in ref. [3] based on the Broadhurst-Kreimer conjecture.
• The computed bases are minimal in depth, meaning no basis with a smaller total depth exists for these weights.
• The basis elements at both weights exhibit a twofold extended structure: each element with two trailing ones corresponds to a Lyndon word in $L_{27}$ or $L_{28}$ via the transformation $Z(m_1, m_2, ..., m_D) \to Z(m_1-1, m_2-1, ..., m_D, 1, 1)$.
• The authors confirm that the pushdown relations predicted by the conjecture—where MZVs can be expressed with reduced depth—coincide with the n-fold extended basis construction, validating the link between pushdowns and extended bases.
• The computational optimizations reduced disk usage from over 800 GB to less than 42 GB, enabling the first successful computation of these bases.
• The results provide strong evidence that the Broadhurst-Kreimer generating functional correctly predicts the structure of MZV bases, including higher-order extensions beyond single pushdowns.