Lecturer(s)


Pech Pavel, prof. RNDr. CSc.

Course content

Afinne varieties, Hilbert's basis theorem, Ideal, Grőbner bases of ideals, Ideal  variety correspondence, Solving polynomial equations, resultant, Buchberger's algorithm, Elimination of variables, Algebraic surfaces, Implicit and parametric equation of a surface, computer methods, The use of surfaces in practice, examples of cubic surfaces and surfaces of a higher order, Surfaces as loci of points.

Learning activities and teaching methods

Monologic (reading, lecture, briefing), Dialogic (discussion, interview, brainstorming), Monitoring

Learning outcomes

Afinne varieties, Hilbert's basis theorem, Ideal, Grőbner bases of ideals, Ideal  variety correspondence, Solving polynomial equations, resultant, Buchberger's algorithm, Elimination of variables, Algebraic surfaces, Implicit and parametric equation of a surface, computer methods, The use of surfaces in practice, examples of cubic surfaces and surfaces of a higher order, Surfaces as loci of points.
Knowledge of basic algorithms of computer algebra and their application.

Prerequisites

Knowledge of geometry in the extent of a univerity basic course of geometry.

Assessment methods and criteria

Oral examination, Written examination
Understanding of basic notions of the theory, knowledge of definitions, theorems and basic proofs. Ability to solve typical problems of a given theory. Written part of the exam is particularly oriented to verification whether a student can solve individual problems which come from a given theory. The aim of the oral part of the exam is to verify student's understanding of the theory and his/her ability to apply it.

Recommended literature


Cox, D., Little, J., O'Shea, D. Ideals, Varieties and Algorithms. Springer Verlag, 1979..

Cox, D., Little, J., O'Shea, D. Using Algebraic Geometry. Birkhäuser, 2005..
