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- W2951569243 abstract "Let $K$ be a field, $mathcal {O}_v$ a valuation ring of $K$ associated to a valuation $v$: $KrightarrowGammacup{infty}$, and ${bf m}_v$ the unique maximal ideal of $mathcal {O}_v$. Consider an ideal $mathcal {I}$ of the free $K$-algebra $Klangle Xrangle =Klangle X_1,...,X_nrangle$ on $X_1,...,X_n$. If ${cal I}$ is generated by a subset $mathcal {G}subset{cal O}_vlangle Xrangle$ which is a monic Grobner basis of ${cal I}$ in $Klangle Xrangle$, where $mathcal {O}_vlangle Xrangle =mathcal{O}_vlangle X_1,...,X_nrangle$ is the free $mathcal{O}_v$-algebra on $X_1,...,X_n$, then the valuation $v$ induces naturally an exhaustive and separated $Gamma$-filtration $F^vA$ for the $K$-algebra $A=Klangle Xrangle /mathcal {I}$, and moreover $mathcal{I}capmathcal{O}_vlangle Xrangle =langlemathcal{G}rangle$ holds in $mathcal{O}_vlangle Xrangle$; it follows that, if furthermore $mathcal{G}notsubset {bf m}_v{O}_vlangle Xrangle$ and $klangle Xrangle /langleoverline{mathcal G}rangle$ is a domain, where $k=mathcal{O}_v/{bf m}_v$ is the residue field of $mathcal{O}_v$, $klangle Xrangle =klangle X_1,...,X_nrangle$ is the free $k$-algebra on $X_1,...,X_n$, and $overline{mathcal G}$ is the image of $mathcal{G}$ under the canonical epimorphism $mathcal{O}_vlangle Xranglerightarrow klangle Xrangle$, then $F^vA$ determines a valuation function $Arightarrow Gammacup{infty}$, and thereby $v$ extends naturally to a valuation function on the (skew-)field $Delta$ of fractions of $A$ provided $Delta$ exists." @default.
- W2951569243 created "2019-06-27" @default.
- W2951569243 creator A5048838044 @default.
- W2951569243 date "2010-11-12" @default.
- W2951569243 modified "2023-09-29" @default.
- W2951569243 title "Valuation Extensions of Algebras Defined by Monic Grobner Bases" @default.
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- W2951569243 doi "https://doi.org/10.48550/arxiv.1011.2860" @default.
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