Syllabus Math 4903/5843 – 001 – Spring 2026

1.1 Polynomials and Affine Space

1.2 Affine Varieties

1.2 Affine Varieties

1.3 Parameterizations of Affine Varieties

1.4 Ideals

1.5 Polynomials of One Variable

2.1 Introduction to Gröbner bases

2.2 Orderings on the Monomials in \(k[x_1, \ldots, x_n]\)

2.3 A division Algorithm in \(k[x_1, \ldots, x_n]\)

2.4 Monomial Ideals and Dickson’s Lemma

2.5 The Hilbert Basis Theorem and Gröbner Bases

2.6 Properties of Gröbner Bases

2.7 Buchberger’s Algorithm

2.8 First Applications of Gröbner Bases 3.1 The Elimination and the Extension Theorem

3.2 The Geometry of Elimination

3.3 Implicitization

Exam 1

4.1 Hilbert’s Nullstellensatz 4.2 Radical Ideals and the Ideal-Variety Correspondence

4.2 Radical Ideals and the Ideal-Variety Correspondence 4.3 Sums, Products, and Intersections of Ideals

4.3 Sums, Products, and Intersections of Ideals 4.4 Zariski Closures, Ideal Quotients, and Saturations

4.4 Zariski Closures, Ideal Quotients, and Saturations

4.4 Zariski Closures, Ideal Quotients, and Saturations 4.5 Irreducible Varieties and Prime Ideals

4.5 Irreducible Varieties and Prime Ideals

4.5 Irreducible Varieties and Prime Ideals 4.6 Decomposition of a Variety into Irreducibles

TBA

TBA

TBA