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.
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Prerequisites:
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 
Additional Notes:
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 nontrivial 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 handson 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 cuttingedge 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: