Thesis Title

Points and Curves in Affine Space

Date of Graduation

Summer 2005

Degree

Master of Science in Mathematics

Department

Mathematics

Committee Chair

Kishor Shah

Keywords

Nullstellensatz, Noether normalization, monomial curve, integral, localization, Groebner basis, affine variety, commutative algebra, algebraic geometry

Subject Categories

Mathematics

Abstract

In the first part of our thesis, we carefully study five versions of the Hilbert’s Null-stellensatz, along with their proofs, as given in David Eisenbud’s Commutative Algebra with a View Toward Algebraic Geometry. We fill in the details for all the proofs. In the second part of our thesis, we explore curves with the parameterization {(t̂m, t̂(m+1),…,t̂(m+k)) │ t ∈ }, were m and k are positive integers. Our Variety Extension Therom determines the defining ideals of these curves under certain conditions. Our Exponent Set Criteria gives sufficient conditions for a polynomial to be zero. This is useful in showing that certain remainders are zero when applying the Division Algorithm. We conjecture a Java program that calculates the reduced Groebner basis for the ideals of these curves.

Copyright

© Daniel R. Kopsas

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