Time and Place: 9:00-9:50am MWF, CB 347
Office Hours: open door policy.
Varieties, and Algorithms (2nd edition), by D. Cox, J. Little and D. O'Shea, ISBN 0-387-94680-2.
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Throughout the semester, the pace of the course will be tailored according to the preparation and needs of the students; plenty of interesting discussions, examples, and illustrations will be provided, although virtually all results are proved. The core of the course introduces such fundamentals as rings of polynomials, varieties, Gröbner bases as well as the basics of Algebraic Geometry such as the Hilbert Nullstellensatz, the connection between ideals and varieties, irreducibility and primary decomposition. Students are assumed to have access to a computer algebra package such as
With the advent of powerful computing resources, it is worth pointing out that concepts and techniques from Algebraic Geometry have recently found applications in areas of Computer Science, Engineering and Robotics. For instance, systematic approaches that use algebraic varieties have been developed to describe the space of possible configurations of mechanical linkages such as robot ``arms." Hence Gröbner bases can be utilized in the solution of the forward and inverse kinematic problem for robots. For these reasons, the course might be of interest to advanced students from the Engineering Department as well.
Course Description: An outline of the course is listed below; in parenthesis we propose a realistic distribution of the weeks for each topic. More details for each of the topics will follow.
Gröbner bases: After having seen how the algebra of the polynomial ring k[x1, ... , xn] and the geometry of affine algebraic varieties are linked, we now study the method of Gröbner bases, which will allow us to solve problems about polynomial ideals in an algorithmic or computational fashion. For instance, does every ideal of k[x1, ... , xn] have a finite generating set (``ideal description problem")?
If we examine in detail the division algorithm in k[x] and the row-reduction (Gaussian elimination) algorithm for systems of linear equations, we see that a notion of ordering of terms in polynomials is a key ingredient of both. A major component of any extension of division and row-reduction to arbitrary polynomials in several variables will be an ordering on the terms in polynomials in k[x1, ... , xn]. We will construct several different examples of orderings, each of which will be useful in different contexts. We will introduce a division algorithm which will allow us to give a (nonconstructive) proof of the existence of a finite generating set for every polynomial ideal (the so called Hilbert Basis Theorem). The treatment will also lead to ideal bases (Gröbner bases) with good properties relative to the division algorithm. We will also address how to construct effectively a Gröbner basis for an ideal. The algorithm of Buchberger is the cornerstone of Computational Algebraic Geometry.
Another problem that we can solve using Gröbner bases is the ``ideal membership problem." We also investigate how the Gröbner bases technique can be applied to ``solve systems of polynomial equations in several variables:" students have already encountered this type of problem when using for instance the Lagrange multiplier method to optimize a function of several variables subject to a certain number of constraints. Another problem deals with the ``implicitization problem:" that is, suppose some (polynomial) parametric equations xi = fi(t1, ..., tm) for i=1, ..., n define an algebraic variety V, how can we find equations in the xi that define V?
Algebra-Geometry dictionary: We are now ready to explore the correspondence between ideals and varieties. We will prove the Hilbert Nullstellensatz, a celebrated theorem which identifies exactly which ideals correspond to varieties. This will allow us to construct a "dictionary" between geometry and algebra. We will define a number of natural algebraic operations (sums, products and intersections) on ideals and study their geometric analogues. In keeping up with the computational emphasis of the course we will develop algorithms to carry out the algebraic operations. We will finally study the more important algebraic and geometric concepts arising out of the Hilbert Basis Theorem: notably, the possibility of decomposing a variety into a union of simpler varieties and the corresponding algebraic notion of writing an ideal as an intersection of simpler ideals.
Functions on a variety: One of the unifying themes of modern mathematics is that in order to understand any class of mathematical objects, one should also study mappings between those objects, and especially the mappings which preserve some properties of interest. For instance, in Linear Algebra after studying vector spaces one studies the properties of linear maps between them. At this stage of the course we will consider mappings between varieties. The algebraic properties of polynomial functions on a variety yield many insights into the geometric properties of the variety itself. We will be lead to the idea of quotient rings and coordinate rings of affine varieties. We will show that two affine varieties are isomorphic if and only if their coordinate rings are isomorphic. We will also introduce the notions of rational mappings between irreducible affine varieties and of birational equivalence of irreducible varieties. Just as isomorphisms of varieties can be detected from the coordinates rings, will show that birational equivalences can be detected from the function fields.
Dimension of a variety: The most important invariant of a linear subspace of a vector space is its dimension. For affine varieties we will carefully define the dimension of any affine variety and show how to compute it. We will show that this notion accords well with what we would expect intuitively. The ideas are introduced via monomial ideals, leading to the Hilbert function and the definition of dimension. Basic properties are explored, including the characterization of dimension in terms of algebraically independent elements. The connection with singularity theory is introduced, and we will also discuss the tangent cone at a singular point of a variety.
Bibliography: The course will closely follow the chapters of the book by Cox, Little and O'Shea. Amplifications of the material will be given when appropriate and depending on the interest of the class. Other relevant (perhaps ambitious) references are listed below: