Weierstrass preparation theorem

Choose a representative of $f$ on a neighbourhood of the closed cylinder $V\times \{\mathrm{\mid }\zeta \mathrm{\mid }\le r\}$ given by [L2]. For each $z^{\mathrm{\prime }}\in V$, let ${\lambda }_1(z^{\mathrm{\prime }}),\dots ,{\lambda }_d(z^{\mathrm{\prime }})$ be the slice zeros in $\mathrm{\mid }\zeta \mathrm{\mid }<r$, counted with multiplicity. By [L3], the elementary symmetric functions $e_1(z^{\mathrm{\prime }}),\dots ,e_d(z^{\mathrm{\prime }})$ of those roots are holomorphic in $z^{\mathrm{\prime }}$. Define $W(z^{\mathrm{\prime }},\zeta ):={\zeta }^d-e_1(z^{\mathrm{\prime }}){\zeta }^{d-1}+\cdots +(-1{)}^d{e}_d(z^{\mathrm{\prime }})\mathrm{.}$ At $z^{\mathrm{\prime }}=0$ all slice roots equal $0$, so $e_j(0)=0$ for every $j$; therefore $W$ is a degree-$d$ Weierstrass polynomial by [L4].

For each fixed $z^{\mathrm{\prime }}\in V$, the polynomial $W(z^{\mathrm{\prime }},\zeta )$ has exactly the zeros ${\lambda }_1(z^{\mathrm{\prime }}),\dots ,{\lambda }_d(z^{\mathrm{\prime }})$ with their multiplicities. Since $f(z^{\mathrm{\prime }},\zeta )$ has the same zero multiset by construction, [L4] shows that at every slice zero $\lambda$ the quotient $f(z^{\mathrm{\prime }},\zeta )\mathrm{/}W(z^{\mathrm{\prime }},\zeta )$ extends holomorphically across $\lambda$; away from those zeros it is holomorphic by [L5]. Thus the slice quotient $q_{z^{\mathrm{\prime }}}(\zeta ):=\frac{f(z^{\mathrm{\prime }},\zeta )}{W(z^{\mathrm{\prime }},\zeta )}$ is holomorphic on $\mathrm{\mid }\zeta \mathrm{\mid }<r$.

Define $u(z^{\mathrm{\prime }},z_m):=\frac{1}{2\pi i}{\int }_{\mathrm{\mid }\zeta \mathrm{\mid }=r}\frac{f(z^{\mathrm{\prime }},\zeta )}{W(z^{\mathrm{\prime }},\zeta )(\zeta -z_m)}d\zeta \mathrm{.}$ Because $W(z^{\mathrm{\prime }},\zeta )\mathrm{\ne }0$ on $\mathrm{\mid }\zeta \mathrm{\mid }=r$, the integrand is continuous on $V\times \{\mathrm{\mid }\zeta \mathrm{\mid }=r\}\times \{\mathrm{\mid }{z}_m\mathrm{\mid }<r\}$ and holomorphic in each parameter variable. By [L6], the resulting function $u$ is separately holomorphic and locally bounded, hence holomorphic on $V\times \{\mathrm{\mid }{z}_m\mathrm{\mid }<r\}$. For fixed $z^{\mathrm{\prime }}$, the one-variable Cauchy formula [L7] applied to the holomorphic slice quotient $q_{z^{\mathrm{\prime }}}$ from step 2.1 gives $u(z^{\mathrm{\prime }},z_m)=q_{z^{\mathrm{\prime }}}(z_m),$ so $f(z^{\mathrm{\prime }},z_m)=u(z^{\mathrm{\prime }},z_m)W(z^{\mathrm{\prime }},z_m)\mathrm{.}$

On the central slice, regularity gives $f(0,\zeta )={\zeta }^{d}h(\zeta )$ with $h(0)\mathrm{\ne }0$, while step 1.1 gives $W(0,\zeta )={\zeta }^d$. Hence step 3.1 yields $u(0,0)=h(0)\mathrm{\ne }0$. By [L1], the germ of $u$ is a unit. Therefore the germs of $u$ and $W$ satisfy $f=uW$ in ${\mathcal{O}}_{m,0}$, which is the required preparation.