Computations about formal multiple zeta spaces defined by binary extended double shuffle relations

The space generated by multiple zeta values (MZVs for short) has been elucidated theoretically and numerically in recent years, but its structure remains mysterious. In this paper, we shed light on a formal space generated by binary analogs of MZVs by computer experiments for unraveling both of the original and formal spaces.

Let $\mathbb{N}$ denote the set of positive integers. The MZV is a real number that belongs to an image of a function (customarily denoted by $\zeta$) whose domain is

$\displaystyle{\bf I}$

$\displaystyle=$

$\displaystyle{\textstyle\bigcup\limits_{r\geq 0}}\{{\bf k}_{r}=(k_{1},k_{2},\ldots,k_{r})\in\mathbb{N}^{r}{\,|\,}k_{1}\geq 2\},$

(1.1)

where ${\bf k}_{0}=\varnothing$ is the empty mult-index and $\zeta(\varnothing)=1$. We call ${\mathrm{w}}({\bf k}_{r})=k_{1}+\cdots+k_{r}$ and ${\mathrm{d}}({\bf k}_{r})=r$ the weight and depth, respectively. The function $\zeta$ has two definitions by the iterated integral and nested summation, which endow the $\mathbb{Q}$-vector space $\mathcal{Z}$ spanned by MZVs with abundant linear relations. Euler [13], who solved the Basel problem $\zeta(2)=\pi^{2}/6$ and advanced the case $r=1$, also studied the case $r=2$.

Zagier [34] conjectured¹¹1 Zagier noted the conjectures were made after many discussions with Drinfel’d, Kontsevich and Goncharov. that $\mathcal{Z}$ is graded by weight and the dimensions of graded pieces are expressed in terms of a Fibonacci-like sequence. Let ${\bf I}_{k}$ be the subset consisting of mult-indices of weight $k$, and let $\mathcal{Z}_{k}$ be the subspace spanned by MZVs in $\zeta({\bf I}_{k})=\{\zeta({\bf k}){\,|\,}{\bf k}\in{\bf I}_{k}\}$. The dimension conjecture is

$\displaystyle\dim_{\mathbb{Q}}\mathcal{Z}_{k}$

$\displaystyle\overset{?}{=}$

$\displaystyle d_{k},$

(1.2)

where $d_{k}=d_{k-2}+d_{k-3}$ $(k\geq 3)$, $d_{0}=d_{2}=1$ and $d_{1}=0$. These integers fit together into the generating series

$\displaystyle\sum_{k\geq 0}d_{k}X^{k}$

$\displaystyle=$

$\displaystyle\frac{1}{1-(X^{2}+X^{3})}.$

(1.3)

The ultimate upper bound theorem (i.e., $\dim_{\mathbb{Q}}\mathcal{Z}_{k}\leq d_{k}$) was established independently by Goncharov [10, 16] and Terasoma [32]. Brown [8] furthermore proved that $\mathcal{Z}_{k}$ is generated by MZVs in $\zeta({\bf I}^{H}_{k})$, where ${\bf I}^{H}_{k}$ is the set of Hoffman mult-indices of weight $k$:

$\displaystyle{\bf I}^{H}_{k}$

$\displaystyle=$

$\displaystyle\{{\bf k}=(k_{1},\ldots,k_{r})\in{\bf I}_{k}{\,|\,}k_{i}\in\{2,3\}\}.$

(1.4)

Hoffman [18] conjectured $\zeta({\bf I}^{H}_{k})$ is a basis of $\mathcal{Z}_{k}$, which would imply the dimension conjecture because the same recurrence relation $|{\bf I}^{H}_{k}|=|{\bf I}^{H}_{k-2}|+|{\bf I}^{H}_{k-3}|$ holds by a simple count of the number of $2$’s and $3$’s. Umezawa [33] also suggested a basis conjecture in terms of iterated log-sine integrals, in which sets of mult-indices different from ${\bf I}^{H}_{k}$ are used. Because of the difficulty to show the independence between MZVs, no non-trivial lower bounds are known.

By the upper bound theorem, it is natural to ask that what sorts of relations are needed to reduce the number of generators of $\mathcal{Z}_{k}$ to $d_{k}$. There are several conjectural candidates: e.g., [11, 14, 17, 22, 23]. In particular, the extended double shuffle (EDS) relations [19, 29] known from early on are often selected for experimentally attacking this question, because they are easier to write down and included in the other candidates except Kawasima’s [23]. Minh and Petitot [28] verified that the class of EDS relations is a right candidate up to weight $10$, Bigotte et al.​ [5] verified it up to weight $12$, Minh et al.​ [27] verified it up to weight $16$,​²²2 This experimental result was announced in their private communication (see [21, Section 1]). Espie et al.​ [12] verified it up to weight $19$, and Kaneko et al.​ [21] verified it up to weight $20$ that seems to be the latest record. The first two experiments are by the Gröbner basis method, and the last three ones are by the vector space (or matrix) method. The fourth one of [12] was executed under modulo rational multiples of powers of $\zeta(2)$, or module $\mathbb{Q}[\zeta(2)]$.

The first purpose of this paper is to improve the record to weight $k=22$. For this, we consider an $\mathbb{F}_{2}$-vector space $\mathcal{Z}_{k}^{\mathfrak{b}}$ instead of the $\mathbb{Q}$-vector space $\mathcal{Z}_{k}$: roughly speaking, $\mathcal{Z}_{k}^{\mathfrak{b}}$ is generated by binary multiple zeta symbols $\zeta^{\mathfrak{b}}({\bf k})$ $({\bf k}\in{\bf I}_{k})$ (binary MZSs for short), where $\zeta^{\mathfrak{b}}({\bf k})$ satisfy binary EDS relations that are obtained from original EDS relations after the modulo $2$ arithmetic to integer coefficients. (Exact definitions of the binary analogs in this section will be stated in the next section.) We will verify $\zeta^{\mathfrak{b}}({\bf I}^{H}_{k})$ is a basis of $\mathcal{Z}_{k}^{\mathfrak{b}}$ and $\dim_{\mathbb{F}_{2}}\mathcal{Z}_{k}^{\mathfrak{b}}=d_{k}$. Our calculation results break the record because $\dim_{\mathbb{Q}}\mathcal{Z}_{k}\leq\dim_{\mathbb{F}_{2}}\mathcal{Z}_{k}^{\mathfrak{b}}$ (as will be mentioned in Section 3). The space $\mathcal{Z}_{k}^{\mathfrak{b}}$ reduces the computation cost since $\mathbb{F}_{2}$ is the binary and simplest finite field. The field $\mathbb{F}_{2}$ makes it easy to apply useful techniques in computer since $\mathbb{F}_{2}$ is compatible with the Boolean datatype: in fact, we will employ a conflict based algorithm discussed in [24] for a fast Gaussian forward elimination.

The second and main purpose is to observe a Pascal triangle pattern in $\mathcal{Z}_{k}^{\mathfrak{b}}$ from the viewpoint of a direct sum decomposition,

$\displaystyle\mathcal{Z}_{k}^{\mathfrak{b}}$

$\displaystyle\cong$

$\displaystyle\overline{\mathcal{Z}}_{k,k-1}^{\mathfrak{b}}\operatorname*{\bigoplus}\cdots\operatorname*{\bigoplus}\overline{\mathcal{Z}}_{k,0}^{\mathfrak{b}},$

(1.5)

where $\overline{\mathcal{Z}}_{k,r}^{\mathfrak{b}}$ are quotient spaces defined by means of depth filtration: the descending chain $\mathcal{Z}_{k,k-1}^{\mathfrak{b}}\supset\cdots\supset\mathcal{Z}_{k,0}^{\mathfrak{b}}$ is used for $\overline{\mathcal{Z}}_{k,r}^{\mathfrak{b}}=\mathcal{Z}_{k,r}^{\mathfrak{b}}/\mathcal{Z}_{k,r-1}^{\mathfrak{b}}$, where $\mathcal{Z}_{k,r}^{\mathfrak{b}}$ are the subspaces spanned by binary MZSs of weight $k$ and depth at most $r$. We define ${\bf I}_{k,r}=\{{\bf k}\in{\bf I}_{k}{\,|\,}{\mathrm{d}}({\bf k})=r\}$ and

$\displaystyle{\bf I}^{H}_{k,r}$

$\displaystyle=$

$\displaystyle{\bf I}^{H}_{k}\cap{\bf I}_{k,r},$

(1.6)

with $d^{\mathfrak{b}}_{k,r}=|{\bf I}^{H}_{k,r}|$. We denote by $\overline{\zeta}^{\mathfrak{b}}({\bf k})$ the canonical image³³3 We use the same notation $\overline{\zeta}^{\mathfrak{b}}$ for all canonical images in the quotient spaces $\overline{\mathcal{Z}}_{k,r}^{\mathfrak{b}}$ $(k>r\geq 0)$. There should be no confusion because the quotient space under consideration is clear from context. of $\zeta^{\mathfrak{b}}({\bf k})$ in $\overline{\mathcal{Z}}_{k,r}^{\mathfrak{b}}$ for any ${\bf k}\in{\bf I}_{k,r}$. Up to weight $k=22$, we will verify $\overline{\zeta}^{\mathfrak{b}}({\bf I}^{H}_{k,r})$ is a basis of $\overline{\mathcal{Z}}_{k,r}^{\mathfrak{b}}$ and $\dim_{\mathbb{F}_{2}}\overline{\mathcal{Z}}_{k,r}^{\mathfrak{b}}=d^{\mathfrak{b}}_{k,r}$. Counting the number of $2$’s and $3$’s implies that the double sequence $(d^{\mathfrak{b}}_{k,r})$ satisfies a recurrence relation with a Pascal triangle pattern: $d^{\mathfrak{b}}_{k,r}=d^{\mathfrak{b}}_{k-2,r-1}+d^{\mathfrak{b}}_{k-3,r-1}$ $(k\geq 3,r\geq 1)$, $d^{\mathfrak{b}}_{0,0}=d^{\mathfrak{b}}_{2,1}=1$ and $d^{\mathfrak{b}}_{k,r}=0$ for other $k$ and $r$, or equivalently,

$\displaystyle\sum_{k,r\geq 0}d^{\mathfrak{b}}_{k,r}X^{k}Y^{r}$

$\displaystyle=$

$\displaystyle\frac{1}{1-(X^{2}+X^{3})Y}.$

(1.7)

More precisely, $d^{\mathfrak{b}}_{k,r}=\binom{r}{k-2r}$ since the integers $P_{r,k}=d^{\mathfrak{b}}_{k+2r,r}$ satisfy the same recurrence relation as the binomial coefficients $\binom{r}{k}$. As expected from (1.5), the formula (1.7) specializes to (1.3) upon $Y=1$.

We also try experiments on parts of EDS relations, ‘$\mathrm{KNT}$’ and ‘$\mathrm{MJPO}$’ relations, which are expected to be alternatives to EDS and actually employed in [21, 27] for verification, respectively. Unlike the case in $\mathcal{Z}_{k}$, those relations do not suffice to give all relations in $\mathcal{Z}_{k}^{\mathfrak{b}}$, but we can find a quasi Fibonacci-like rule in dimensions of spaces defined by $\mathrm{MJPO}$ relations.

The idea of the depth filtration in (1.5) was conceived by Broadhurst and Kreimer [7] to propose a refinement of the dimension conjecture. Their conjecture indicates two interesting facts in the $\mathbb{Q}$-vector spaces of MZVs graded by both weight and depth: (i) modular forms influence the structure through quotient spaces $\overline{\mathcal{Z}}_{k,r}$ defined by the $\mathbb{Q}$-version of (1.5); and (ii) the Hoffman values $\zeta({\bf k})$ $({\bf k}\in{\bf I}^{H}_{k})$ are irrelevant to the structure in the sense that most of the values vanish in the graded pieces of same depth. In terms of the generating series, the conjecture is

$\displaystyle\sum_{k,r\geq 0}\dim\overline{\mathcal{Z}}_{k,r}X^{k}Y^{r}$

$\displaystyle\overset{?}{=}$

$\displaystyle\frac{1+E(X)Y}{1-O(X)Y+S(X)Y^{2}(1-Y^{2})},$

(1.8)

where $E(X)=X^{2}/(1-X^{2})$, $O(X)=X^{3}/(1-X^{2})$ and $S(X)=X^{12}/(1-X^{4})(1-X^{6})$, and $S(X)$ is the generating series of the dimensions of the vector spaces of cusp forms on the full modular group. Specific examples for $r=2$ are given in [15] and a modern formulation is discussed in [9] (see also [31]). However our computational results suggest the following when we adopt $\mathbb{F}_{2}$ as the scalar field instead of $\mathbb{Q}$: (i) the influence of modular forms disappears; but (ii) the Hoffman symbols $\zeta^{\mathfrak{b}}({\bf k})$ $({\bf k}\in{\bf I}^{H}_{k})$ remain as basis elements with a Pascal triangle pattern.

It should be noted that the Broadhurst-Kreimer conjecture has two equivalent formulations of vector and algebra (see [19, Appendix]). The equivalence requires $\mathbb{Q}[\zeta(2)]$ is isomorphic to the polynomial ring in one variable over $\mathbb{Q}$. The isomorphy does not hold when $\mathbb{F}_{2}$ is the scalar field as will be mentioned in the final section, and we will consider only the vector formulation in this paper.

It should also be noted that Blümlein et al.​ [6] provided a data mine for not only MZVs but also Euler sums by experiments to Broadhurst-Kreimer type conjectures, in which it was verified that the union of EDS and duality relations suffices to reduce the number of generators of $\mathcal{Z}_{k}$ to $d_{k}$ up to weight $22$: it was also verified up to $24$ by using modular arithmetic, and up to $26$ and more with an additional conjecture and limited depths. The duality relations, which are obtained by the integral definition of MZVs and a change of variables, are very useful to compute because they can bring down the size of relations by about half. It has not been proved yet that the EDS relations include the duality relations, although the inclusion is expected to be true conjecturally: in other words, we have not succeeded in understanding the duality of MZVs algebraically. The experimental approaches of [6] and ours differ in the use of the duality relations.

The organization of this paper is as follows. In Section 2, we state exact definitions of the binary MZVs $\zeta^{\mathfrak{b}}({\bf k})$, the formal multiple zeta spaces $\mathcal{Z}_{k}^{\mathfrak{b}}$ and the quotient spaces $\overline{\mathcal{Z}}_{k,r}^{\mathfrak{b}}$. We report our computational results in Section 3, and explain how our computer programs produce the results in Section 4. The programs are available at the open-source site GitHub.⁴⁴4https://github.com/machide-tomoyan/BMZS-calculator Section 5 is devoted to problems about formal multiple zeta spaces which arise from the computational results. In Appendix, we describe an essential algorithm in our experiments, which employs a conflict based search and speeds up the Gaussian forward elimination under certain conditions.

The computer only assists us in showing Proposition 4.1 by Gaussian elimination. The dimension conjecture (1.2) is true if we can theoretically show Proposition 4.1 for all weights $k$.

The formal multiple zeta space $\mathcal{Z}_{k}^{\mathfrak{b}}$ of weight $k$ is briefly defined by

$\displaystyle\mathcal{Z}_{k}^{\mathfrak{b}}$

$\displaystyle=$

$\displaystyle\frac{\langle\eta^{\mathfrak{b}}({\bf k}){\,|\,}{\bf k}\in{\bf I}_{k}\rangle_{\mathbb{F}_{2}}}{\text{ \{binary EDS relations\} }},$

(2.1)

where $\eta^{\mathfrak{b}}({\bf k})$ are indeterminates. That is, $\mathcal{Z}_{k}^{\mathfrak{b}}$ is an $\mathbb{F}_{2}$-vector space generated by formal symbols $\zeta^{\mathfrak{b}}({\bf k})\equiv\eta^{\mathfrak{b}}({\bf k})$ that satisfy binary variations of the EDS relations. Eight equivalent statements are given in [19, Theorem 2] for the EDS relations. In this paper, we choice the statement (v) in the theorem because the relations are all $\mathbb{Z}$-linear and fewer in number.

To define (2.1) exactly, we require the algebraic setup by Hoffman [18] which allows us the steady handling of two products, the shuffle $\mathcyr{sh}$ and stuffle $*$: the latter is also called harmonic or quasi-shuffle. Let $\mathfrak{H}$ be the polynomial ring $\mathbb{Q}\langle x,y\rangle$ in the two non-commutative variables $x$ and $y$. We call each variable a letter, and a monomial in the variables a word. The shuffle product $\mathcyr{sh}$ is a $\mathbb{Q}$-bilinear product on $\mathfrak{H}$, which satisfies $w=w\;\mathcyr{sh}\;1=1\;\mathcyr{sh}\;w$ and

$\displaystyle au\;\mathcyr{sh}\;bv$

$\displaystyle=$

$\displaystyle a(u\;\mathcyr{sh}\;bv)+b(au\;\mathcyr{sh}\;v)$

(2.2)

for any words $u,v,w\in\mathfrak{H}$ and letters $a,b\in\{x,y\}$. Let $z_{k}$ denote a word $x^{k-1}y$ for any $k\geq 1$, and let $\mathfrak{H}^{1}$ be the polynomial ring $\mathbb{Q}\langle z_{1},z_{2},\ldots\rangle$, or equivalently, the subring $\mathbb{Q}+\mathfrak{H}y$ in $\mathfrak{H}$. The stuffle product $*$ is a $\mathbb{Q}$-bilinear product on $\mathfrak{H}^{1}$, which satisfies $w=w\,*\,1=1\,*\,w$ and

$\displaystyle z_{i}u\,*\,z_{j}v$

$\displaystyle=$

$\displaystyle z_{i}(u\,*\,z_{j}v)+z_{j}(z_{i}u\,*\,v)+z_{i+j}(u\,*\,v)$

(2.3)

for any words $u,v,w\in\mathfrak{H}$ and integers $i,j\geq 1$. By induction on the lengths of words, both products are commutative and associative, and both $\mathfrak{H}^{1}_{\mathcyr{sh}}=(\mathfrak{H}^{1},\mathcyr{sh})$ and $\mathfrak{H}^{1}_{*}=(\mathfrak{H}^{1},*)$ are commutative $\mathbb{Q}$-algebras. We notice $\mathfrak{H}_{\mathcyr{sh}}=(\mathfrak{H},\mathcyr{sh})$ is a parent space of $\mathfrak{H}^{1}_{\mathcyr{sh}}$. Let $\mathfrak{H}^{0}=\mathbb{Q}+x\mathfrak{H}y=\langle z_{{\bf k}}{\,|\,}{\bf k}\in{\bf I}\rangle_{\mathbb{Q}}$, where $z_{{\bf k}}=z_{k_{1}}\cdots z_{k_{r}}$ and $z_{\varnothing}=1$. Both $\mathfrak{H}^{0}_{\mathcyr{sh}}=(\mathfrak{H}^{0},\mathcyr{sh})$ and $\mathfrak{H}^{0}_{*}=(\mathfrak{H}^{0},*)$ are subalgebras since $\mathfrak{H}^{0}$ is closed under $\mathcyr{sh}$ and $*$. The pair $(\mathfrak{H}^{1},\mathfrak{H}^{0})$ of spaces satisfies the polynomial ring property in one variable: the former is freely generated by $y$ over the latter on each of $\mathcyr{sh}$ and $*$. We thus have

$\displaystyle\mathfrak{H}^{1}_{\mathcyr{sh}}\,\simeq\,\mathfrak{H}^{0}_{\mathcyr{sh}}[y],\qquad\mathfrak{H}^{1}_{*}\,\simeq\,\mathfrak{H}^{0}_{*}[y].$

(2.4)

See [30] and [18] for proofs of (2.4), respectively.

We introduce the EDS relations stated in [19, Theorem 2(v)]. Let $\mathrm{reg}_{\mathcyr{sh}}$ denote a homomorphism from $\mathfrak{H}^{1}_{\mathcyr{sh}}$ to $\mathfrak{H}^{0}_{\mathcyr{sh}}$, which is defined by taking the constant term with respect to $y$ in the first isomorphism of (2.4):⁵⁵5 The homomorphism $\mathrm{reg}_{*}$ of stuffle type exists as well, but it is intractable because EDS relations of that type are not always $\mathbb{Z}$-linear: see [19] (or [2, 20]) for details.

$\displaystyle\mathrm{reg}_{\mathcyr{sh}}:\mathfrak{H}^{1}_{\mathcyr{sh}}\,\ni\,w\,=\,\sum_{i=0}^{m}w_{i}\;\mathcyr{sh}\;y^{\mathcyr{sh}i}\quad\mapsto\quad w_{0}\,\in\,\mathfrak{H}^{0}.$

(2.5)

Let ${\bf\widehat{I}}_{k}={\bf I}_{k}\cup\{(\underset{k}{\underbrace{1,\ldots,1}})\}$, and let

$\displaystyle\widehat{{\bf PI}}_{k}$

$\displaystyle=$

$\displaystyle{\textstyle\bigcup\limits_{i,j\geq 0\atop(i+j=k)}}{\bf\widehat{I}}_{i}\times{\bf I}_{j}.$

For any pair $({\bf k},{\bf l})$ of mult-indices in $\widehat{{\bf PI}}_{k}$, we define

$\displaystyle\mathsf{ds}({\bf k},{\bf l})$

$\displaystyle:=$

$\displaystyle\mathrm{reg}_{\mathcyr{sh}}(z_{{\bf k}}\,*\,z_{{\bf l}})-\mathrm{reg}_{\mathcyr{sh}}(z_{{\bf k}}\;\mathcyr{sh}\;z_{{\bf l}})\,\in\,\mathfrak{H}^{0}.$

(2.6)

The objective EDS relations of weight $k$ are stated as

$\displaystyle Z(\mathsf{ds}({\bf k},{\bf l}))$

$\displaystyle=$

$\displaystyle 0\qquad(({\bf k},{\bf l})\in\widehat{{\bf PI}}_{k}),$

(2.7)

where $Z:\mathfrak{H}^{0}\to\mathbb{R}$ is the $\mathbb{Q}$-linear map (or evaluation map) defined by $Z(z_{{\bf k}})=\zeta({\bf k})$ $({\bf k}\in{\bf I})$. We have by (2.5)

	$\displaystyle\mathrm{reg}_{\mathcyr{sh}}(w)$	$\displaystyle=$	$\displaystyle w\qquad(w\in\mathfrak{H}^{0}),$
	$\displaystyle\rule{0.0pt}{15.0pt}\mathrm{reg}_{\mathcyr{sh}}(y^{m}\;\mathcyr{sh}\;z_{{\bf m}})$	$\displaystyle=$	$\displaystyle 0\qquad(m>0,{\bf m}\in{\bf I}).$

We can thus divide (2.7) into two parts:

	$\displaystyle Z(z_{{\bf k}}\,*\,z_{{\bf l}})-Z(z_{{\bf k}}\;\mathcyr{sh}\;z_{{\bf l}})$	$\displaystyle=$	$\displaystyle 0\qquad(({\bf k},{\bf l})\in{\bf PI}_{k}),$		(2.8)
	$\displaystyle\rule{0.0pt}{15.0pt}Z(\mathrm{reg}_{\mathcyr{sh}}(y^{m}\,*\,z_{{\bf m}}))$	$\displaystyle=$	$\displaystyle 0\qquad(0<m<k-1,{\bf m}\in{\bf I}_{k-m}),$		(2.9)

where ${\bf PI}_{k}={\textstyle\bigcup_{i,j\geq 0\atop(i+j=k)}}{\bf I}_{i}\times{\bf I}_{j}$. The relations in (2.8) are called the finite double shuffle (FDS) relations, because MZVs are defined by $\zeta(k_{1},\ldots,k_{r})=\sum_{m_{1}>\cdots>m_{r}>0}1/m_{1}^{k_{1}}\cdots m_{r}^{k_{r}}$ and finite (or convergent) at ${\bf k}\in{\bf I}$. The FDS relations do not suffice to give all relations of MZVs. For instance, we can not obtain any relation in weight $3$, in particular, the simplest formula $\zeta(2,1)=\zeta(3)$. Therefore the relations in (2.9) are essential to the EDS conjecture.

A little more notions are required for (2.1), which are analogs of the notions mentioned above in $\mathbb{Z}$-module and $\mathbb{F}_{2}$-vector. Let $\mathfrak{H}^{\mathbb{Z}}$ denote the subring $\mathbb{Z}\langle x,y\rangle$ in $\mathbb{Q}\langle x,y\rangle$. We set

$\displaystyle\mathfrak{H}^{\mathbb{Z},0}\,=\,\langle z_{{\bf k}}{\,|\,}{\bf k}\in{\bf I}\rangle_{\mathbb{Z}},\qquad\mathcal{H}^{\mathfrak{b}}\,=\,\langle\eta^{\mathfrak{b}}({\bf k}){\,|\,}{\bf k}\in{\bf I}\rangle_{\mathbb{F}_{2}},$

to define a canonical map from $\mathfrak{H}^{\mathbb{Z},0}$ to $\mathcal{H}^{\mathfrak{b}}$ which is induced by modulo $2$ arithmetic:

$\displaystyle\mathrm{can}^{\mathfrak{b}}:\mathfrak{H}^{\mathbb{Z},0}\,\ni\,w\,=\,\sum_{{\bf k}\in{\bf I}}c_{{\bf k}}z_{{\bf k}}\quad\mapsto\quad\sum_{{\bf k}\in{\bf I}}(c_{{\bf k}}\,\mathrm{mod}\ 2)\eta^{\mathfrak{b}}({\bf k})\,\in\,\mathcal{H}^{\mathfrak{b}}.$

(2.10)

For any pair $({\bf k},{\bf l})\in{\bf PI}_{k}$, the elements $z_{{\bf k}}\,*\,z_{{\bf l}}$ and $z_{{\bf k}}\;\mathcyr{sh}\;z_{{\bf l}}$ belong to $\mathfrak{H}^{\mathbb{Z},0}$, and the element $\mathrm{can}^{\mathfrak{b}}(\mathsf{ds}({\bf k},{\bf l}))$ is well-defined. For $0<m<k-1$ and ${\bf m}\in{\bf I}_{k-m}$, $y^{m}\,*\,z_{{\bf m}}$ belongs to $\langle y^{n}z_{{\bf n}}{\,|\,}n\geq 0,{\bf n}\in{\bf I}\rangle_{\mathbb{Z}}$, and $\mathrm{can}^{\mathfrak{b}}(\mathrm{reg}_{\mathcyr{sh}}(y^{m}\,*\,z_{{\bf m}}))$ is well-defined if

$\displaystyle\mathrm{reg}_{\mathcyr{sh}}(y^{n}z_{{\bf n}})$

$\displaystyle\in$

$\displaystyle\mathfrak{H}^{\mathbb{Z},0}\qquad(n>0,{\bf n}\in{\bf I}),$

which holds by [19, Proposition 8] (see (4.7) below). Consequently,

$\displaystyle\mathcal{E}^{\mathfrak{b}}_{k}$

$\displaystyle:=$

$\displaystyle\langle\mathrm{can}^{\mathfrak{b}}(\mathsf{ds}({\bf k},{\bf l})){\,|\,}({\bf k},{\bf l})\in\widehat{{\bf PI}}_{k}\rangle_{\mathbb{F}_{2}}\,\subset\,\mathcal{H}^{\mathfrak{b}}_{k}$

is well-defined.

We are in a position to define (2.1).

Let $H^{\mathfrak{b}}$ denote the natural homomorphism from $\operatorname*{\bigoplus}_{k\geq 0}\mathcal{H}^{\mathfrak{b}}_{k}$ to $\operatorname*{\bigoplus}_{k\geq 0}\mathcal{Z}_{k}^{\mathfrak{b}}$: each component is the canonical map of (2.11). We define the binary evaluation map by $Z^{\mathfrak{b}}=H^{\mathfrak{b}}\circ\mathrm{can}^{\mathfrak{b}}$. The binary EDS relations of weight $k$ are then stated as

$\displaystyle Z^{\mathfrak{b}}(\mathsf{ds}({\bf k},{\bf l}))$

$\displaystyle=$

$\displaystyle 0\qquad(({\bf k},{\bf l})\in\widehat{{\bf PI}}_{k}).$

(2.12)

We list some examples of the original and binary EDS relations for weights $k\leq 4$ in Table 1.

Let $\mathcal{Z}_{k,r}^{\mathfrak{b}}$ denote the vector subspace $\langle\zeta^{\mathfrak{b}}({\bf k}){\,|\,}{\bf k}\in{\bf I}_{k},{\mathrm{d}}({\bf k})\leq r\rangle_{\mathbb{F}_{2}}$ as introduced in the first section. We end this section with the definition of the graded pieces satisfying the direct sum decomposition (1.5).

We report our computational results. How we obtain them will be explained in the next section.

We begin with a typical result related to (1.2).

The EDS conjecture states that, for every weight $k$, the relations in (2.7) suffice to reduce the number of generators of $\mathcal{Z}_{k}$ to $d_{k}$:

$\displaystyle\dim_{\mathbb{Q}}\mathcal{E}_{k}$

$\displaystyle\geq$

$\displaystyle 2^{k-2}-d_{k},$

(3.2)

where $\mathcal{E}_{k}=\langle\mathsf{ds}({\bf k},{\bf l}){\,|\,}({\bf k},{\bf l})\in\widehat{{\bf PI}}_{k}\rangle_{\mathbb{Q}}$. This can be confirmed by Experiment 3.1, as follows. We denote by $\mathcal{E}^{\mathbb{Z}}_{k}=\langle\mathsf{ds}({\bf k},{\bf l}){\,|\,}({\bf k},{\bf l})\in\widehat{{\bf PI}}_{k}\rangle_{\mathbb{Z}}$ the $\mathbb{Z}$-module counterpart of $\mathcal{E}_{k}$. Since $\mathbb{Q}$ is the field of fractions of $\mathbb{Z}$ and $\mathrm{can}^{\mathfrak{b}}$ is a surjective homomorphism from $\mathcal{E}^{\mathbb{Z}}_{k}$ to $\mathcal{E}^{\mathfrak{b}}_{k}$,

$\displaystyle\dim_{\mathbb{Q}}\mathcal{E}_{k}$

$\displaystyle=$

$\displaystyle\mathrm{rank}_{\mathbb{Z}}\,\mathcal{E}^{\mathbb{Z}}_{k}\,\geq\,\dim_{\mathbb{F}_{2}}\mathcal{E}^{\mathfrak{b}}_{k},$

which, together with (3.1), proves (3.2) for $k\leq 22$.

We recall $d^{\mathfrak{b}}_{k,r}=\binom{r}{k-2r}$ that is the number of the Hoffman mult-indices of weight $k$ and depth $r$. We define $\mathcal{H}^{\mathfrak{b}}_{k,r}=\langle\eta^{\mathfrak{b}}({\bf k}){\,|\,}{\bf k}\in{\bf I}_{k},{\mathrm{d}}({\bf k})\leq r\rangle_{\mathbb{F}_{2}}\subset\mathcal{H}^{\mathfrak{b}}_{k}$, and

$\displaystyle\overline{\mathcal{E}}^{\mathfrak{b}}_{k,r}$

$\displaystyle=$

$\displaystyle(\mathcal{H}^{\mathfrak{b}}_{k,r}\cap\mathcal{E}^{\mathfrak{b}}_{k})/\mathcal{H}^{\mathfrak{b}}_{k,r-1}.$

The main result is a refinement of Experiment 3.1. Taking the sum for $r=1,\ldots,k-1$ in (3.3) induces (3.1) because of (1.5): note that $\overline{\mathcal{Z}}_{k,0}^{\mathfrak{b}}=\{0\}$ unless $k=0$.

The first equality in (3.3) is by the isomorphism theorems. In fact, we have

	$\displaystyle\mathcal{Z}_{k,r}^{\mathfrak{b}}/\mathcal{Z}_{k,r-1}^{\mathfrak{b}}$	$\displaystyle\simeq$	$\displaystyle\raisebox{4.0pt}[0.0pt][0.0pt]{$\mathcal{H}^{\mathfrak{b}}_{k,r}/(\mathcal{H}^{\mathfrak{b}}_{k,r}\cap\mathcal{E}^{\mathfrak{b}}_{k})$}\Big{/}\raisebox{-4.0pt}[0.0pt][0.0pt]{$\mathcal{H}^{\mathfrak{b}}_{k,r-1}/(\mathcal{H}^{\mathfrak{b}}_{k,r-1}\cap\mathcal{E}^{\mathfrak{b}}_{k})$}$		(3.4)
		$\displaystyle\simeq$	$\displaystyle\rule{0.0pt}{15.0pt}\raisebox{4.0pt}[0.0pt][0.0pt]{$\mathcal{H}^{\mathfrak{b}}_{k,r}$}\Big{/}\raisebox{-4.0pt}[0.0pt][0.0pt]{$(\mathcal{H}^{\mathfrak{b}}_{k,r-1}+\mathcal{H}^{\mathfrak{b}}_{k,r}\cap\mathcal{E}^{\mathfrak{b}}_{k})$}$
		$\displaystyle\simeq$	$\displaystyle\rule{0.0pt}{15.0pt}\raisebox{4.0pt}[0.0pt][0.0pt]{$\mathcal{H}^{\mathfrak{b}}_{k,r}/\mathcal{H}^{\mathfrak{b}}_{k,r-1}$}\Big{/}\raisebox{-4.0pt}[0.0pt][0.0pt]{$(\mathcal{H}^{\mathfrak{b}}_{k,r}\cap\mathcal{E}^{\mathfrak{b}}_{k})/\mathcal{H}^{\mathfrak{b}}_{k,r-1}$},$

and

$\displaystyle\overline{\mathcal{Z}}_{k,r}^{\mathfrak{b}}$

$\displaystyle\simeq$

$\displaystyle\raisebox{4.0pt}[0.0pt][0.0pt]{$\mathcal{H}^{\mathfrak{b}}_{k,r}/\mathcal{H}^{\mathfrak{b}}_{k,r-1}$}\Big{/}\raisebox{-4.0pt}[0.0pt][0.0pt]{$\overline{\mathcal{E}}^{\mathfrak{b}}_{k,r}$}.$

Since $\binom{k-2}{r-1}=|\mathcal{H}^{\mathfrak{b}}_{k,r}/\mathcal{H}^{\mathfrak{b}}_{k,r-1}|$ by counting the number of the mult-indices of weight $k$ and depth $r$, we obtain the desired equality.

We demonstrate the numbers $d^{\mathfrak{b}}_{k,r}$ for $k\leq 22$ in Table 2. They are expressed in terms of binomial coefficients, and we can observe a (shifted) Pascal triangle pattern: the column $r=0$ has the sequence $(1)$ from the row $k=0$, the column $r=1$ has $(1,1)$ from $k=2$, the column $r=2$ has $(1,2,1)$ from $k=4$, the column $r=3$ has $(1,3,3,1)$ from $k=6$, and so on. For comparison, the dimensions of ${\overline{\mathcal{Z}}_{k,r}}$ conjectured in (1.8) are listed in Table 3.

Refinements of the EDS conjecture have been proposed. Minh et al.​ [27] conjectured that a part of the EDS relations obtained from

$\displaystyle\widehat{{\bf PI}}_{k}^{\mathrm{MJPO}}$

$\displaystyle=$

$\displaystyle{\bf PI}_{k}\cup({\bf\widehat{I}}_{1}\times{\bf I}_{k-1})$

(3.5)

is a right candidate, and verified it up to $k=16$. The relations

$\displaystyle Z(\mathsf{ds}({\bf k},{\bf l}))$

$\displaystyle=$

$\displaystyle 0\qquad(({\bf k},{\bf l})\in{\bf\widehat{I}}_{1}\times{\bf I}_{k-1})$

are known as Hoffman’s relations ([18]), and their conjecture says that FDS relations and Hoffman’s relations suffice to give all relations among MZVs. Kaneko et al.​ [21] conjectured the above relations are too much, i.e., a smaller part obtained from

$\displaystyle\widehat{{\bf PI}}_{k}^{\mathrm{KNT}}$

$\displaystyle=$

$\displaystyle(\{(3),(2,1)\}\times{\bf I}_{k-3})\cup(\{(2)\}\times{\bf I}_{k-2})\cup({\bf\widehat{I}}_{1}\times{\bf I}_{k-1})$

(3.6)

is a right candidate. They verified it up to $k=20$.

In the space $\mathcal{Z}_{k}^{\mathfrak{b}}$, neither the relations obtained from (3.5) nor those obtained from (3.6) suffice to give all relations among binary MZSs.