MAGIC112: Computational Algebra and Geometry

Since the 1960s, a new field of mathematics has been steadily growing in significance. Broadly describable as computational algebra, this field grew out of algorithms for computing Gröbner bases: that is, for finding explicit generators for ideals. Some notable successes have stimulated a wide interest in the area, from disciplines such as engineering and robotics through to algebraic geometry and cryptography.

Topics in abstract algebra will be approached from the perspective of what can be explicitly computed, with an emphasis on applications to algebraic geometry. Although the emphasis is on computation, no use of a computer (and certainly no programming) will be required, although you can choose to make use of them. We use the term "computational" in the more formal sense to mean those things that can be explicitly calculated via algorithms, in contrast to much of your experience in pure mathematics to date, which has avoided the issue of direct computation.

Familiarity with the basics of algebra: rings, fields, etc.

(1)  Introduction to Rings and ideals.
Introduces the concept of a commutative ring and its ideals, focusing on polynomial rings k[x_1,...,x_n]. Introduces the key questions this course addresses:
 - Are all ideals I \subset k[x_1,...,x_n] in a polynomial ring finitely generated?
 - If I \subset k[x_1,...,x_n] is known to be finitely generated, can we write down a set of generators?
 - Is this set of generators unique?
 - Is there any systematic way of determining whether a polynomial f is in I?

(2)  Introduction to Algebraic Geometry.
Introduces affine space A^n, and the varieties in affine space V(I) generated by systems of polynomial equations (and ideals). Introduces the "inverse" process of moving from an affine variety to an ideal I(V), and asks the key question of when I and V truly are inverses of each other.

(3)  Polynomials in One Variable.
Considers polynomials in one variable and the corresponding ideals. The Division Algorithm is proven and used to show that all ideals I are principal. The Euclidian Algorithm is proven and used to derive a generator for I, and hence uniqueness.

(4)  Polynomials in k[x_1,...,x_n].
Considers the more difficult case of polynomials in more than one variable. Introduces the concept of total order, and the standard monomial orders: lexicographic order, graded lexicographic order, and reverse graded lexicographic order. States the Division Algorithm and considers its many weaknesses in comparison to the case of polynomials in k[x].

(5)  Hilbert Basis Theorem.
Introduces monomial ideals, proves Dickson’s Lemma concerning finite generation, and as a corollary proves the Hilbert Basis Theorem.

(6)  Gröbner Bases and Buchberger’s Algorithm.
Defines the concept of a Gröbner basis for an ideal I in a polynomial ring, motivated as a way of addressing the failures of the Division Algorithm in k[x_1,...,x_n]. Defines the S-polynomial, and uses this to state and prove Buchberger’s Criterion, which tells us when a set of generators of I is a Gröbner basis. States and proves Buchberger’s Algorithm, allowing us to turn any set of generators into a Gröbner basis. Concludes by noticing that Gröbner bases are not unique.

(7)  Minimal and Reduced Gröbner Bases.
Introduces the notion of minimal and reduces Gröbner bases. Proves the main result, that the reduced Gröbnerbasis of I is unique, thus resolving all of the shortcomings of the Division Algorithm in many variables.

(8)  Elimination Theory.
Now we start applying the concept of reduced Gröbner basis. We begin with Elimination Theory, which lets us eliminate a variable from a system of polynomial equations. Through induction, this allows us to find the solutions to systems of polynomial equations that we otherwise wouldn’t be able to solve. Here the key definition is the l-th elimination ideal, which we use to state and prove the Elimination Theorem, and the Extension Theorem.

(9)  Projecting Affine Varieties.
We relate Elimination Theory to the geometric idea of projecting affine varieties. We state the Closure Theorem, however the proof cannot be given until later in the course. Also mentions the concept of Zariski Topology.

(10) The Nullstellensatz.
Introduces the definition of a radical ideal \sqrt{I}, telling us exactly when V and I are inverses of each other.

(11) Radical Algorithms.
The proof of the Nullstellensatz motivates an algorithm for determining when f \in I. We also state and prove a method for computing the radical \sqrt{I} of I.

(12) Prime Ideals and Irreducible Varieties.
Introduces the geometric definition of an irreducible affine variety, and the algebraic concept of a prime ideal. Proves how the two ideas are connected. Proves various results on prime ideals, including some sufficient conditions for when a variety is irreducible. Introduces the definition of a maximal ideal, and considers the geometric interpretation.

(13) Zariski Closure.
Uses the Nullstellensatz to proves the Closure Theorem. Explores the geometric interpretation of closure, and the algebraic concept of the colon ideal I:J. States and proves a theorem relating the two: V(I:J) = \overline{V(I)\V(J)}. Describes and proves an algorithm for computing this in many cases, allowing us to write a variety as a union of irreducible components.