Research of the cancellation problem in affine algebraic geometry

根据参考文献中的论文摘要：(1)给出了特征为p>2$的域$k$上的几何正射平面曲线的纯不可分$k$-形式的代数结构定理。(2)设$R$是一个具有商域$K$和剩馀域$K$的离散估值环。对于有限生成的整数闭域$ a $ / $R$，当一般纤维$ Aotimes_Rk $为一元多项式环或一元一元洛朗多项式环时，给出了约简$k$-代数$(Aotimes_Rk)_{rm red}$的显式代数结构。(3)设$V$是一个赋值环，它具有残域$k$和商域$k$。设$K(x,y)$是在$K$上由多项式方程$f(x,y)=0$定义的一元代数函数域，设$V_{xy}$是$K(x,y)$的赋值环，其残差域$k_{xy}$。假设$V_{xy}$支配$V$，且$k_{xy}/k$是超越域扩展。则$k_{xy}$是一元代数函数域/ $k$。我们考虑$k_{xy}$的$k$-代数结构。特别地，当$k$是特征为零的代数闭域时，当$y^2=x^m+lambda_1x+lambda_0$对于$0<m$和$lambda_0,lambda_1in k$，我们将显式地给出$k_{xy}=k(z,w)$的定义方程$g(z,w)$。

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 $lambda_0,lambda_1in K$.