 | Seminario Matematico e Fisico di Milano |
| Olivier Debarre
Sorbonne Université - Université de Paris
When can solutions of polynomial equations be algebraically parametrized?
www.mate.polimi.it/smf/index.php?settore=home&id_link=25
Lunedì 01 Febbraio 2021, ore 10:30
https://polimi-it.zoom.us/j/83674264668 |
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Abstract
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The description of all the solutions of the equation $x^2+y^2=z^2$ in integral numbers (a.k.a. Pythagorean triples) is a very ancient problem: a Babylonian clay tablet from about 1800BC may contain some solutions, Pythagoras (about 500BC) seems to have known one infinite family of solutions, and so did Plato... This gives a first example of a rational variety: the rational points on the circle with equation $x^2+y^2=1$ can be algebraically parametrized by one rational parameter. More generally, one says that a variety, defined by a system of polynomial equations, is rational if its points (the solutions of the system) can be algebraically parametrized, in a one-to-one fashion, by independent parameters. I will begin with easy standard examples, then explain and apply some (not-so-recent) techniques that can be used to prove that some varieties (such as the set of rational solutions of the equation $x^3+y^3+z^3+t^3=1$) are not rational. |
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