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Abstract:Let $k$ be any algebraically closed field in any characteristic,
let $R$ be any regular local ring such that $R$ contains $k$ as a subring,
the residue field of $R$ is isomorphic to $k$ as $k$-algebras and $\dim R\geq 1$,
let $P$ be any parameter system of $R$ and let $z\in P$. We consider any $\phi\in R$ with $\phi\neq 0$.
In our main theorem we assume several conditions depending on $P$, $z$ and Newton polyhedrons.
By our assumptions the normal fan $\Sigma$ of the Newton polyhedron $\Gamma_+(P,\phi)$ of $\phi$ over $P$
has simple structure and we can make a special regular subdivision $\Sigma^*$ of $\Sigma$ called
an upward subdivision, starting from a regular cone with dimension equal to $\dim R$
and repeating star subdivisions with center in a regular cone of dimension two.
Let $X$ and $\sigma:X\rightarrow\Spec(R)$ denote the toric variety over $\Spec(R)$
and the toric morphism associated with $\Sigma^*$. $X$ is irreducible and smooth and $\sigma$ is a composition of
finite blowing-ups with center in a closed irreducible smooth subscheme of codimension two.
We consider any closed point $a\in X$ such that $\sigma(a)$ is the unique closed point of $\Spec(R)$,
the local ring $\mathcal{O}_{X,a}$ of $X$ at $a$ and the morphism $\sigma^*: R \rightarrow\mathcal{O}_{X,a}$ of
local $k$-algebras induced by $\sigma$. We show that our numerical invariant of
$\sigma^*(\phi)\in \mathcal{O}_{X,a}$ measuring the badness of the singularity is strictly less than
the same invariant of $\phi\in R$ and the singularity $\phi$ is strictly improved by $\sigma$.
We notice that this result opens a way toward the theory of resolution of singularities in arbitrary characteristic.
We add several submain theorems to make bridges toward it and to show that our assumptions of the main theorem are
not strong.
By these results we can show that in a mathematical game with two players A and B related to the resolution of
singularities of $\phi$, the player A can always win the game after finite steps.
It follows ``the local uniformization theorem in arbitrary characteristic and in arbitrary dimension''.
Keywords: resolution of singularities, blowing-up, normal crossing, smooth, regular local ring, Newton polyhedron, toric theory, convex polyhedral cone, regular cone
Subject class: Primary 14E15; Secondary 32S45, 52A20
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