(in Polish) Algorytmy w algebrze 1000-1M18AA
The course is designed mainly for students interested in commutative algebra and algebraic geometry, but should be also useful for these interested in algebraic topology and homological algebra. We will study how to apply computer algebra systems to access and understand certain properties of rings, ideals and varieties, how to interpret these results and, in some cases, make computations more efficient. We assume good knowledge of basic algebra (as in the Algebra I and Commutative algebra courses), especially ring theory. No experience in programming is required.
Tentative course plan:
1 Groebner bases (3-4 lectures): Buchberger's algorithm, examples and applications, in particular elimination theory and computing homomorphism kernel.
2 Resolutions, syzygies, deformations (4-6 lectures), Betti numbers, many examples (including ones coming from geometry), Hilbert function and Hilbert polynomial, resolutions in families.
3 Properties of rings and ideals (2-3 lectures): normality and Serre's criterion, localisation, saturation, homogenisation.
4 Additional topics (selection):
(a) Finite groups: classification, representations, character tables (GAP)
(b) Invariants of finite group actions (Singular)
(c) Khovanskii bases - Groebner-type bases for algebras
(d) Basic toric geometry - algorithms for cones and polytopes (Macaulay, Singular, Polymake)
(e) Basic toric geometry - varieties, morphisms (Magma)
(f) Computing tropicalisations
(g) Symmetric polynomials
Type of course
Prerequisites
Prerequisites (description)
Learning outcomes
Students understand most important theorem and algorithms on which computer algebra systems are based. Moreover, they know how to approach solving problems, in particular analysing examples coming from algebra and (algebraic) geometry using computer algebra systems.
Assessment criteria
Oral exam.
Bibliography
"Ideals, Varieties, and Algorithms", David A. Cox, John B. Little, Donal O'Shea
"Computational Commutative Algebra", M. Kreuzer, L. Robbiano
"A Singular Introduction to Commutative Algebra", G.-M. Greuel and G. Pfister
"Computations in Algebraic Geometry with Macaulay2", D. Eisenbud, D.R. Grayson, M. Stillman, B. Sturmfels (eds)
"Commutative Algebra", D. Eisenbud
"Geometry of Syzygies", D. Eisenbud
"Commutative Algebra", H. Matsumura
"Groebner Bases and Convex Polytopes", B. Sturmfels
"Combinatorial Commutative Algebra", E. Miller, B. Sturmfels
manuals of Macaulay2, Singular, Magma, GAP
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:
Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system: