Valuation ideals in polynomial rings

Seidenberg, A.

doi:10.1090/s0002-9947-1945-0012270-3

Cited by 5 publications

(3 citation statements)

References 4 publications

(4 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The notation we establish will be valid simultaneously for algebraic number fields and algebraic function fields of one variable. Let Oo be either the ring 7 If p is a maximal ideal in o, let Op be the ring of quotients of o with respect to p. If po = pn0o, denote by 0p0 the quotient ring of o0 with respect to p0. Op and 0p" are local rings and, moreover, the completion o*0 of 0p0 is a subring and subspace of the completion o* of Op.…”

Section: Daniel Gorensteinmentioning

confidence: 99%

“…A generalization of a theorem of Seidenberg. Seidenberg [7 ] has given a detailed analysis of the valuation ideals in a polynomial ring in two independent variables over an algebraically closed ground field. As a special case of his theory one can obtain the corresponding analysis for the valuation ideals in a nonhomogeneous coordinate ring of a plane algebraic curve (over an algebraically closed ground field).…”

Section: Theoremmentioning

confidence: 99%

See 1 more Smart Citation

An arithmetic theory of adjoint plane curves

Gorenstein¹

1952

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

Section: Daniel Gorensteinmentioning

confidence: 99%

Section: Theoremmentioning

confidence: 99%

An arithmetic theory of adjoint plane curves

Gorenstein¹

1952

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…A generalization of a theorem of Seidenberg. Seidenberg [7 ] has given a detailed analysis of the valuation ideals in a polynomial ring in two independent variables over an algebraically closed ground field. As a special DANIEL GORENSTEIN [May case of his theory one can obtain the corresponding analysis for the valuation ideals in a nonhomogeneous coordinate ring of a plane algebraic curve (over an algebraically closed ground field).…”

Section: Daniel Gorensteinmentioning

confidence: 99%

An Arithmetic Theory of Adjoint Plane Curves

Gorenstein¹

1952

Transactions of the American Mathematical Society

View full text Add to dashboard Cite

Introduction.In classical algebraic geometry the adjoint curves to an irreducible plane curve are an essential tool in the study of the geometry on the curve. In this paper we shall give an algebro-arithmetic development of the theory of adjoint curves, and shall extend the classical results to irreducible plane curves with arbitrary singularities defined over arbitrary ground fields. Our definition of the adjoint condition at a given singular point of the curve is stated in terms of the conductor between the local ring of the point and its integral closure. The fundamental properties of the adjoint curves are then derived from corresponding properties of the conductor.The single deepest and most important property of the adjoint curves is that, on a curve of order m, the adjoint curves of order m -3 cut out the complete canonical series. This property is equivalent to the fact that the

show abstract