Section 2 Course Information
Subsection 2.1 Course Description
This course is, first and foremost, an introduction to classical algebraic geometry and commutative algebra. The central is to build a dictionary between the geometry of affine space and the algebra of polynomial rings, culminating in a proof of Hilbert’s Nullstellensatz. We will utilize modern computational tools, such as SageMath and Gröbner bases, to simplify computations and visualize the geometry of systems of polynomial equations.
Subsection 2.2 Course Prerequisites
A grade of C or better in MATH 3083 (formerly MATH 308) and 4083 (formerly MATH 408) or permission from the instructor.
Subsection 2.3 Student Learning Outcomes
By the end of this course, students will be able to:
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Explain and apply the correspondence between classical affine varieties and polynomial ideals.
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State and apply fundamental results of polynomial rings, including Hilbert’s Basis Theorem and Nullstellensatz.
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Perform computations in multivariate polynomial rings—including polynomial division, ideal membership tests, and Gröbner basis computations using computer algebra systems such as SageMath.
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Use Gröbner bases to analyze polynomial ideals and solve systems of polynomial equations.
Subsection 2.4 Course Topics
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Polynomial rings and Ideals
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Affine Varieties
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Monomial orderings
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Division Algorithm for Polynomials
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The Hilbert Basis Theorem
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Gröbner bases and Buchberger’s Algorithm
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Elimination Theory
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Parameterization and Implicitization
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Envelopes
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Resultants
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Hilbert’s Nullstellensatz
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The Ideal-Variety correspondence
Subsection 2.5 Instructional Methods
This course is offered as a face-to-face course. Learning will be facilitated through traditional lecture, group work/activities, homework, and in-class assessments.
