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I am studying a theorem by Ishi in which she proves there is a resolution of the singularity of a hypersurface defined on de affine space considered as the toric variety given by the fan $\sum_i \mathbb{R}_{\geq 0}e_i$ where $(e_i)$ is the canonical base of $\mathbb{R}^{n+1}$.

We know there is a resolution of singularities of a toric variety, given by the unimodular subdivision of its fan. Ishi proves this subdivision also gives a resolution of a singularity of an hypersurface defined on the space, but I am confused with the proof.

At one point she does this: If $X$ is the hf (hypersurface) defined by a polynomial $f(x_0,\dots,x_n)$ and $(y_0,\dots,y_n)$ are the coordinates of the affine toric variety defined by a maximal dimension $\sigma$ within the aforementioned subdivision, then the polynomial factors as $$f(x_0,\dots,x_n)=y_0^m\left( g_0(y_1,\dots, y_n)+y_0\,g_1(y_0,\dots,y_n) \right) $$

I can't get where this factorization comes from. I guess it has something to do with the blowup defined by the stellar subdivision, but I still don't get it.

Hope anyone can shed some light upon this. Thank you.

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