Which cubics admit rational parametrization?
Ivan Cheltsov
The University of Edinburgh
https://www.maths.ed.ac.uk/cheltsov/
Date(s) : 14/03/2025 iCal
16h00 - 17h00
Every day we deal with geometric objects defined by algebraic equations (circles, parabolas, hyperbolas, splines, spheres, hyperboloids, etc). In many applications, we have to parametrize them using the simplest possible functions – rational functions in several variables. To find such parametrization maybe tricky. This is a very classical problem – rational parametrization of a circle has been found by Pythagoras when he found Pythagoras triples, and the same approach gives explicit rational parametrization of a sphere or any geometric object given by one quadratic equation. In more complicated cases, the problem can be very difficult. Moreover, quite often rational parametrization does not exist – there are many algebraic objects that do not admit rational parametrization. For example, majority of plane cubic curves cannot be parametrized by rational functions (the proof for Fermat cubic curves follows from Euler’s proof of Fermat’s Last Theorem for exponent 3). In my talk, I will focus on the existence of a rational parametrization of cubics – geometric objects defined by one equation of degree 3 (cubic curves, cubic surfaces, cubic 3-folds, etc).
Emplacement
Saint-Charles - FRUMAM (2ème étage)
Catégories
