Raíces de Polinomios Aleatorios
Primer Cuatrimestre 2012
Importante
- El día 20 de marzo no habrá clases. Las clases comienzan el jueves 22 de marzo.
- Horarios: martes de 14 a 16hs y jueves de 13 a 15hs.
- Fecha de inicio: jueves 22 de marzo.
- El curso será dictado en español.
- Duración: 24 horas en total (16 de teorica, 8 de consultas) a lo largo de 1 mes.
- Correlatividad: Probablidades y Estadística. Se recomienda también Análisis Complejo.
- Puntaje: Esta materia otorga 1 (un) punto para Licenciatura y Doctorado.
Docente, horarios y aulas
|
Teórico
|
Ma: 14 a 16 Ju: 13 a 15 |
Andre Galligo | Aula: 14 | Pab: I |
| Consultas | A convenir |
Resumen
This is an introductory course at a graduate level which consists of 16 hours divided in 4 weeks.
The subject is the study of real and compex roots of random univariate polynomials and of characteristic polynomials of random matrices.
This subject meets interests of researchers in complexity theory, in computer algebra, in theoretical physics, and in probability theory.
We will give necessary definitions and properties, sketch the proofs of the main results, refereeing to the important bibliography. We will point out some further reading and directions of research.
Contenido del curso:
Week 1 : Univariate real polynomials
- Localisation of real and complex roots.
- Budan-Fourier theorem and virtual roots.
- Budan tables and continuity properties.
- Complex roots near the real axis.
Week 2 : Random polynomials
- A very brief review of probability theory.
- Classical Gaussian distributions of coefficients.
- Density of the number of real roots
- k-points correlations.
- Extension to non Gaussian distributions.
- Budan tables of random polynomials.
Week 3 : Random matrices
- Classical Gaussian distributions of coefficients.
- Distribution of eigenvalues.
- k-points correlations formulas.
- Extension to non Gaussian random matrices.
Week 4 : Applications and related topics
- Brownian motion.
- Determinantal point processes.
- Roots of systems of random polynomials.
Bibliografía
- Krick, Teresa: Polinomios y Raíces (Descargar)
- Rahman, Q.I and Schmeisser,G: Analytic theory of polynomials, Oxford Univ. press. (2002). Chapter 10.
- Bochnack, J. and Coste, M. and Roy, M-F.: Real Algebraic Geometry. Springer (1998). First two chapters.
- Galligo, A: Budan Tables of Real Univariate Polynomials, Preprint 2011, and proceedings ISSAC2011.
- K Farahmand:Topics in Random Polynomials, Research Notes in Mathematics , Series, (1998).
- Edelman, A. and Kostlan, E. How Many Zeros of a Random Polynomial are Real? Bull. Amer. Math. Soc. 32, 1-37, 1995.
- Bleher,P and Xiaojun Di : Correlations Between Zeros of a Random Polynomial . Journal of Statistical Physics, Vol. 88, Nos. 1/2, 1997.
