s}{\vec r};\vep)=\sum r_i Z(\wvec{\vec s-\vec e _i}{\vec r};\vep),
\mlabel{eq:zdiff}$$ where $\vec e_i, 1\leq i\leq k+1$ is the $i$-th unit vector in $\ZZ ^{k+1}$. Each term on the right hand side is in $\CC\{\{\vep,\vep^{-1}\}[\ln(-\vep)]$ by Case 1. So by Lemma , $Z(\wvec{\vec s}{\vec r};\vep)$ is in $\CC[T][[\vep,\vep^{-1}]$. The inductive step follows from Eq. () and the induction assumption.

Renormalized directional MZVs
-----------------------------

Combining Eq. (), Theorem  and Eq. (), we obtain an algebra homomorphism $$\uni{Z}: \calh_\frakM \to \CC\{\{\vep,\vep^{-1}\}[\log (-\vep)] \ola{u} \CC[T][[\vep,\vep^{-1}],\ \wvec{\vec{s}}{\vec{r}} \mapsto
    u\big(Z(\wvec{\vec{s}}{\vec{r}};\vep)\big).
\mlabel{eq:zmap}$$ In the same way, $\uni{Z}$ restricts to an algebra homomorphism $$\uni{Z}: \calh_{\frakM^-} \to \CC[[\vep,\vep^{-1}].$$ For any commutative $\bfk$-algebra $K$, $K[[\vep,\vep^{-1}]$ is a Rota-Baxter algebra of weight -1 with the Rota-Baxter operator $P$ to be the projection to $\vep^{-1} K [\vep^{-1}]$: $$P\big ( \sum_{n\geq N} \alpha_k \vep^k\big )=\sum_{k\leq -1}\alpha_k \vep^k.
 \mlabel{eq:prb}$$ This can be directly verified as with the well-known case of $\CC[[\vep,\vep^{-1}]$ in .

Thus we can apply the Algebraic Birkhoff Decomposition in Theorem  and obtain

We have $$\uni{Z}=\uni{Z}_-^{-1} \star \uni{Z}_+$$ and the map $\uni{Z}_+: \calh_\frakM\to \CC[T][[\vep]]$ is an algebra homomorphism which restricts to an algebra homomorphism $\uni{Z}_+: \calh_{\frakM^-}\to \CC[[\vep]]$.

Because of Corollary , the following definition is valid.

For $\vec{s}=(s_1,\cdots,s_k)\in \ZZ^k$ and $\vec{r}=(r_1,\cdots,r_k)\in \RR_{>0}^k$, define 