# Department of Mathematics Syllabus

This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.

## MAT 165: Mathematics and Computers

Approved: 2010-10-01 (revised 2013-06-01, )
Suggested Textbook: (actual textbook varies by instructor; check your instructor)
Ideals, Varieties, and Algorithms, 3rd edition by Cox, Little and Oshea; Springer; ISBN-13 # 978-0387356501; \$40.00. Also recommended: Introduction to Maple, 3rd edition; by Andre Heck; Springer; ISBN-13 # 978-0387002309; \$47.
Search by ISBN on Amazon: 978-0387356501
Prerequisites:
(MAT 127A or MAT 108 or MAT 114 or MAT 115A or MAT 145); (MAT 022A or MAT 027A or MAT 067 or BIS 027A).
Suggested Schedule:
 Lecture(s) Sections Comments/Topics Week 1 Motivation, introduction to MAPLE and other symbolic computation systems. Exact representation of numbers and polynomials. The algebra of polynomials, the remainder theorem and roots. GCD of polynomials and the Euclidean algorithm. Week 2 The fundamental theorem of algebra. Real and rational roots of polynomials, Sturm sequences and Descartes' rule of signs. Week 3 Ideals and varieties Week 4 Term orders and the multivariate division algorithm Weeks 5 & 6 Groebner bases and Buchberger's algorithm Weeks 7 & 8 Solving systems of multivariate polynomial equations. Hilbert's Nullstellensatz and unsolvable systems. Weeks 9 & 10 Applications in science and engineering
The basic goal of the course is to introduce undergraduates to algebraic and symbolic computation. This course bears roughly the same relationship to algebra that MAT 128ABC (numerical computation) does to analysis, and addresses a complementary set of problems in applied mathematics. We consider problems for which an exact answer is both feasible and desirable. A second goal is to introduce computers as useful tools for mathematical research.
Learning Goals:

This course provides a first introduction to computational mathematics, symbolic computation, and computer generated/verified proofs in algebra, analysis and geometry. The underlying theme of the course is learning how to solve systems of polynomial equations in several variables. This serves as the foundation for the following :

1. Introduce students to basic concepts of algebraic geometry, giving them an opportunity to do some non-trivial pure mathematics. In this regard, the course goes as far as Hilbert's Nullstellensatz, whose proof is done in complete detail. For many undergraduates, this can be a first introduction to rigorous proofs and mathematical thinking.

2. Equal emphasis is put upon theory and hands-on computations with software like MAPLE, Mathematica. This gives undergraduates their first brush with algebraic and symbolic computation. The course also demonstrates how computers are enormously useful tools for mathematical research. Thus, students gain familiarity with computer technology, software, and algorithmic processes necessary in quantitative analysis and mathematical modeling.

3. Students are exposed to cutting-edge research in real world applications of algebraic geometry, e.g., Robotics, Optimization problems in science and engineering. This also serves as an attraction for students from engineering and the natural sciences, and helps to build bridges to other departments on the UC Davis campus.

Assessment:
The course is assessed by having two midterms, a final exam and 3 computer projects in MAPLE/Mathematica. There are also regular homeworks posted for consolidating (and sometimes extending upon) the material done in class.