Hilbert polynomials and the Bezout Theorem

Throughout, (R, m) will denote a d-dimensional CohenMacaulay (CM for short) local ring having an infinite residue field and I an m-primary ideal in R. Recall that an ideal J I is said to be a reduction of I if Ir+1 = JIr for some r 0, and a reduction J of I is called a minimal reduction of I if J is generated by a system of parameters. The concepts of reduction and minimal reduction were first introduced by Northcott and Rees12. If J is a reduction of I, define the reduction number of I with respect to J, denoted by rj(I), to be min {r 0 Ir+1 = JIr}. The reduction number of I is defined as r(I) = min {rj(I)J is a minimal reduction of I}. The reduction number r(I) is said to be independent if r(I) = rj(I) for every minimal reduction J of I.

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Computing Polynomials of the Ramanujan tn Class Invariants

Canadian Mathematical Bulletin ◽

10.4153/cmb-2009-058-6 ◽

2009 ◽

Vol 52 (4) ◽

pp. 583-597 ◽

Cited By ~ 6

Author(s):

Elisavet Konstantinou ◽

Aristides Kontogeorgis

Keyword(s):

Quadratic Field ◽

Imaginary Quadratic Field ◽

Class Field ◽

Minimal Polynomials ◽

Hilbert Polynomials ◽

Reciprocity Law ◽

Class Invariants ◽

Hilbert Class Field

AbstractWe compute the minimal polynomials of the Ramanujan values tn, where n ≡ 11 mod 24, using the Shimura reciprocity law. These polynomials can be used for defining the Hilbert class field of the imaginary quadratic field and have much smaller coefficients than the Hilbert polynomials.

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Rational cuspidal curves in projective surfaces. Topological versus algebraic obstructions

International Journal of Mathematics ◽

10.1142/s0129167x17501063 ◽

2017 ◽

Vol 28 (14) ◽

pp. 1750106

Author(s):

Maciej Borodzik

Keyword(s):

Floer Homology ◽

Singular Points ◽

The Other ◽

Heegaard Floer Homology ◽

Heegaard Floer ◽

Bezout Theorem ◽

Projective Surfaces

We study rational cuspidal curves in projective surfaces. We specify two criteria obstructing possible configurations of singular points that may occur on such curves. One criterion generalizes the result of Fernandez de Bobadilla, Luengo, Melle–Hernandez and Némethi and is based on the Bézout theorem. The other one is a generalization of the result obtained by Livingston and the author and relies on Ozsváth–Szabó inequalities for [Formula: see text]-invariants in Heegaard Floer homology. We show by means of explicit calculations that the two approaches give very similar obstructions.

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