Study on subrings of polynomial rings

研究交换环R上n元多项式环的子环A的下列问题：(1)找出环R上A有限生成的条件；(2)找出A是R上的多项式环或A^&lt；[r]-&gt；的条件。对于第一个问题，我与Amartya K.duta博士合作，研究了R是离散赋值环且n=1的情况，给出了R上A的闭纤维有限生成的条件。针对这一结果，我研究了唯一因式分解整环R上一元多项式环的Noether子环A，给出了A在R上有限生成的条件，并证明了在此条件下，A是多项式环.对于问题(2)，我研究了唯一因式分解整环R上的忠实平坦整环A，使得R上A的通称和余维1的纤维是一元多项式环.对于问题(2)，我和富山大学的T·Asanuma教授研究了如下问题：设R是K上一元代数函数域K(x，y)的赋值环，V是V占优的一元代数函数域K(x，y)的赋值环。找出V的余域的代数结构。对于这个问题，我们研究了K(x，y)是由y^2=x^n+ax+b定义的超椭圆函数域的情况，证明了在控制R的K(x，y)的赋值环V中，至多存在一个V，使得V的余域不是R的余域上的一元有理函数域，并由此确定了V的余域的定义方程.

The purpose of the study is to consider the following problem on subrings A of a polynomial ring in n-variables over a commutative ring R :(1) Find conditions for A to be finitely generated over R.(2) Find conditions for A to be a polynomial ring or an A^<[r]->fibration over R.For the first problem, in collaboration with Dr.Amartya K.Dutta, I investigated the case where R is a discrete valuation ring and n=1, and gave a condition for the closed fiber of A over R to be finitely generated. In connection with this result, I studied Noetherian subrings A of a polynomial ring in one variable over a unique factorization domain R, and gave a condition for A to be finitely generated over R. Furthermore I proved that, under this condition, A is a polynomial ring.For the problem (2), I investigated a faithfully flat integral domain A over a unique factorization domain R such that generic and codimension one fibers of A over R are polynomial rings in one variable. I proved that such A is a direct limit of certain algebras, and using this result I gave a condition for A to be a polynomial ring.Concerning the problem (2), I studied the following problem with Professor T.Asanuma (Toyama University) :Let R be a valuation ring with quotient field K and let V be a valuation ring of an algebraic function field K(x,y) in one variable over K such that V dominates R. Find out the algebraic structure of the residue field of V.For this problem, we investigated the case where K(x,y) is a hyperelliptic function field defined by y^2=x^n+ax+b, and proved that among the valuation rings V of K(x,y) dominating R, there exists at most one V such that the residue field of V is not a rational function field in one variable over the residue field of R. Furthermore, for such V, we determined the defining equation of the residue field of V.