Algebra, Geometry and Calculus

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

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). 

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".

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.

Hourly volumes:

            CM : 21h

            TD : 21h

            TP : 0

            Land : 0