ECTS
5 credits
Component
Faculty of Science
Description
This course develops different aspects of effective calculus in algebra (linear algebra on Euclidean rings, resultants,...) and introduces the first elements of algebraic geometry.
Objectives
To master the "effective" aspects of the algebra notions studied in the first semester, and to prepare students for the modeling option of the Aggregation of Mathematics exam; the UE includes computer work in SageMath (formal calculation).
Necessary pre-requisites
Usual Fourier series in L2(R/Z), reduction of endomorphisms, Jordan, the first semester course "Algebra 1".
Recommended prerequisites: Fourier series in L2(R/Z), reduction of endomorphisms, Jordan, the first semester course "Algebra 1".
Syllabus
1. Effective linear algebra over Euclidean rings: Hermite and Smith normal forms, Frobenius reduction, invariant factors of a matrix. Adapted basis theorem and applications.
2. Fast multiplication of polynomials: complexity calculations and usual multiplication, Karatsuba algorithm, Fourier transform of a finite abelian group, discrete Fourier transform for polynomials, fast Fourier transform algorithm.
3. Resultant: Sylvester matrix, elimination methods, expression in terms of roots, discriminant.
4. Introduction to algebraic geometry: affine space, studies of plane curves (e.g. quadrics), the nullstellensatz, Zariski topology.
5. Groebner basics: monomial orders, Buchberger algorithm and elimination algorithms.
Additional information
Hourly volumes:
CM : 21h
TD : 21h
TP : 0
Land : 0