Algebra, Geometry and Calculus

This course develops various aspects of effective calculus in algebra (linear algebra over Euclidean rings, resultants, etc.) and introduces the first elements of algebraic geometry.

Master the "effective" aspects of the algebra concepts studied in the first semester, and prepare students for the modeling option in the agrégation de Mathématiques competitive examination; the UE includes SageMath computer practical work (formal calculation). 

Usual Fourier series in ^L2(R/Z), reduction of endomorphisms, Jordan, the first-semester course "Algebra 1".

Recommended prerequisites: Usual Fourier series in ^L2(R/Z), reduction of endomorphisms, Jordan, first-semester course "Algebra 1".

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. Result: Sylvester matrix, elimination methods, root expression, discriminant.

4. Introduction to algebraic geometry: affine space, studies of plane curves (e.g. quadrics), the nullstellensatz, Zariski topology.

5. Groebner bases: monomial orders, Buchberger algorithm and elimination algorithms.

Hourly volumes :

            CM: 21h

            TD: 21h

            TP: 0

            Land: 0