Toric resolutions of strongly mixed weighted homogeneous polynomial germs of type $J_{10}^{-}$

Let $f(\boldsymbol{z},\bar{\boldsymbol{z}})$ be a mixed function on $U\ (\subset\mathbb{C}^{n})$. We say $\boldsymbol{a}=(a_{1},\dots,a_{n})\in U$ is a mixed critical point (or a mixed singular point) of $f$ if the rank of the differential map

is less than $2$. We say $\boldsymbol{a}\in U$ is a mixed regular point of $f$ if it is not a mixed critical point of $f$.

Let $(f,\boldsymbol{0})$ be the germ of a mixed function $f$ at $\boldsymbol{0}\in\mathbb{C}^{n}$. For $I\subset\{1,\dots,n\}$, $f^{I}$ denotes the restriction of $f$ on the coordinate subspace $\mathbb{C}^{I}:=\{\boldsymbol{z}\,|\,z_{j}=0,\,j\notin I\}$. A mixed function germ $(f,\boldsymbol{0})$ is called convenient if $f^{I}\not\equiv 0$ for every $I\subset\{1,2,\cdots,n\}$ with $I\neq\emptyset$.

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  A mixed polynomial $f(\boldsymbol{z},\bar{\boldsymbol{z}})=\sum_{\nu,\mu}c_{\nu,\mu}\boldsymbol{z}^{\nu}\bar{\boldsymbol{z}}^{\mu}$ is called radially weighted homogeneous if there exists a weight vector $P={}^{t}(p_{1},\dots,p_{n})\ (\neq\boldsymbol{0})\in\mathbb{Z}_{+}^{n}$ and a positive integer $d_{r}\ (>0)$ such that

  $$c_{\nu,\mu}\neq 0\Longrightarrow P(\nu+\mu)=\sum_{i=1}^{n}p_{i}(\nu_{i}+\mu_{i})=d_{r}.$$

  We call $d_{r}$ the radial degree of $f$, and define ${\rm{rdeg}\/}_{P}f:=d_{r}.$

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  A mixed polynomial $f(\boldsymbol{z},\bar{\boldsymbol{z}})=\sum_{\nu,\mu}c_{\nu,\mu}\boldsymbol{z}^{\nu}\bar{\boldsymbol{z}}^{\mu}$ is called polar weighted homogeneous if there exists a weight vector $Q={}^{t}(q_{1},\dots,q_{n})\ (\neq\boldsymbol{0})\in\mathbb{Z}^{n}$ and an integer $d_{p}$ ($>0,\ 0\ \text{or}\ <0$) such that

  $$c_{\nu,\mu}\neq 0\Longrightarrow Q(\nu-\mu)=\sum_{i=1}^{n}q_{i}(\nu_{i}-\mu_{i})=d_{p}.$$

  We call $d_{p}$ the polar degree of $f$, and define ${\rm{pdeg}\/}_{Q}f:=d_{p}.$

We say a mixed polynomial $f(\boldsymbol{z},\bar{\boldsymbol{z}})$ is a mixed weighted homogeneous polynomial if it is both radially and polar weighted homogeneous. Here, the corresponding weight vectors $P$ and $Q$ are possibly different. We say a mixed weighted homogeneous polynomial $f(\boldsymbol{z},\bar{\boldsymbol{z}})$ is a strongly mixed weighted homogeneous polynomial if $f$ is radially and polar weighted homogeneous with respect to the same weight vector $P$. Furthermore, a mixed weighted homogeneous polynomial $f$ is called a strongly polar positive mixed weighted homogeneous polynomial with respect to a weight vector $P$ if $f$ is radially and polar weighted homogeneous with respect to the same weight vector $P$ and ${\rm{pdeg}\/}_{P}f>0$.

Let $(f,\boldsymbol{0})$ be a mixed function germ at $\boldsymbol{0}\in\mathbb{C}^{n}$. The germ $(f,\boldsymbol{0})$ of a mixed function $f(\boldsymbol{z},\bar{\boldsymbol{z}})$ at $\boldsymbol{0}\in\mathbb{C}^{n}$ is defined to be of strongly polar positive mixed weighted homogeneous face type if for every compact face ¹¹1 In [8], a convenient (Definition 2) mixed function germ $(f(\boldsymbol{z},\bar{\boldsymbol{z}}),\boldsymbol{0})$ is defined to be of strongly polar positive mixed weighted homogeneous face type if the face function $f_{\Delta}(\boldsymbol{z},\bar{\boldsymbol{z}})$ is a strongly polar positive mixed weighted homogeneous polynomial for every $(n-1)$-dimensional face. Besed on this definition, Proposition 10 of [8] asserts that the face function $f_{\Delta(P)}$ is also a strongly polar positive mixed weighted homogeneous polynomial with respect to $P$ for every weight vector $P$ when $(f,\boldsymbol{0})$ is a convenient mixed function germ of strongly polar positive mixed weighted homogeneous face type in the sense of [8]. $\Delta$, the face function $f_{\Delta}(\boldsymbol{z},\bar{\boldsymbol{z}})$ is a strongly polar positive mixed weighted homogeneous polynomial (Definition 7) with respect to some strictly positive weight vector $P$ with $\Delta=\Delta(P)$.

Let $(f,\boldsymbol{0})$ be the germ of a mixed function $f$ at $\boldsymbol{0}\in\mathbb{C}^{n}$.

1. (1)

   We say $(f,\boldsymbol{0})$ is Newton non-degenerate over a compact face $\Delta$ if $0$ is not a mixed critical value of the face function $f_{\Delta}:{\mathbb{C}^{*}}^{n}\to\mathbb{C}$. (In particular, if $f_{\Delta}^{-1}(0)\cap{\mathbb{C}^{*}}^{n}=\emptyset$, then $0$ is not a mixed critical value of the face function $f_{\Delta}:{\mathbb{C}^{*}}^{n}\to\mathbb{C}$.)

2. (2)

   Let $\Delta$ be a compact face with $\dim\Delta\geq 1$. We say $(f,\boldsymbol{0})$ is strongly Newton non-degenerate over $\Delta$ if the face function $f_{\Delta}:{\mathbb{C}^{*}}^{n}\to\mathbb{C}$ has no mixed critical points and $f_{\Delta}:{\mathbb{C}^{*}}^{n}\to\mathbb{C}$ is surjective onto $\mathbb{C}$.

3. (3)

   Let $\Delta$ be a compact face with $\dim\Delta=0$, that is, $\Delta$ is a vertex of $\Gamma_{+}(f)$. We say $(f,\boldsymbol{0})$ is strongly Newton non-degenerate over $\Delta$ if the face function $f_{\Delta}:{\mathbb{C}^{*}}^{n}\to\mathbb{C}$ has no mixed critical points.

We say the germ $(f,\boldsymbol{0})$ of a mixed function $f$ at $\boldsymbol{0}\in\mathbb{C}^{n}$ is Newton non-degenerate (respectively, strongly Newton non-degenerate) if $(f,\boldsymbol{0})$ is Newton non-degenerate (respectively, strongly Newton non-degenerate) over every compact face $\Delta$.

We set $E_{j}:={}^{t}(\underbrace{0,\dots,0,1}_{j},0,\dots,0)\ (\in\mathbb{Z}^{n})$ for every $j\ (=1,\dots,n)$, and

where $J:=\{j_{1},\dots,j_{k}\}\ (\subsetneqq\{1,\dots,n\})$. Let $(f,\boldsymbol{0})$ be the germ of a mixed function $f$ at $\boldsymbol{0}\in\mathbb{C}^{n}$. For $I\subset\{1,\dots,n\}$, $f^{I}$ denotes the restriction of $f$ on the coordinate subspace $\mathbb{C}^{I}:=\{\boldsymbol{z}\,|\,z_{j}=0,\,j\notin I\}$. Let $\Sigma^{*}$ be a regular simplicial cone subdivision which is admissible for $\Gamma^{*}(f)$. We say $\Sigma^{*}$ is convenient if for every $I$ with $f^{I}\not\equiv 0$, the cone $E_{I^{c}}$ is contained in $\Sigma^{*}$.