Study on subrings of polynomial rings

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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.