2005 Fiscal Year Final Research Report Summary
Research of the cancellation problem in affine algebraic geometry
| Project/Area Number |
16540016
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | University of Toyama |
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Principal Investigator |
ASANUMA Teruo University of Toyama, Faculty of Science, Professor, 理学部, 教授 (50115127)
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| Co-Investigator(Kenkyū-buntansha) |
ONODA Nobuharu University of Fukui, Faculty of Engineering, Professor, 工学部, 教授 (40169347)
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| Project Period (FY) |
2004 – 2005
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| Keywords | affine algebraic geometry / algebraic curve / polynomial ring / valuation ring / cancellation problem / algebraic function field |
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| Research Abstract |
From the abstracts of papers in the references :
(1)We give an algebraic structure theorem of purely inseparable $k$-forms of geometrically normal affine plane curves over a field $k$ of characteristic $p>2$.
(2)Let $R$ be a discrete valuation ring with quotient field $K$ and residue field $k$. For a finitely generated integrally closed domain $A$ over $R$, we give an explicit algebraic structure of the reduced $k$-algebra $(Aotimes_Rk)_{rm red}$ when the generic fibre $Aotimes_RK$ is a polynomial ring or a Laurent polynomial ring in one variable over $K$.
(3)Let $V$ be a valuation ring with residue field $k$ and quotient field $K$. Let $K(x,y)$ be an algebraic function field in one variable over $K$ defined by a polynomial equation $f(x,y)=0$ and let $V_{xy}$ be a valuation ring of $K(x,y)$ with residue field $k_{xy}$. Suppose that $V_{xy}$ dominates $V$ and $k_{xy}/k$ is a transcendental field extension. Then $k_{xy}$ is an algebraic function field in one variable over $k$. We consider a $k$-algebraic structure of $k_{xy}$. In particular when $k$ is an algebraic closed field of characteristic zero, we will give a defining equation $g(z,w)$ of such $k_{xy}=k(z,w)$ explicitly when $y^2=x^m+lambda_1x+lambda_0$ for $0<m$ and $lambda_0,lambda_1in K$.
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