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Optimal Parametrizations of Algebraic Curves

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Prof. Rafael Sendra (University Madrid), 22 April 2010, 2:30 p.m., RISC seminar room
When Apr 22, 2010
from 02:30 PM to 04:00 PM
Where RISC seminar room
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Optimal Parametrizations of Algebraic Curves

In this talk, after a brief introduction to basic concepts on algebraic curves, we plan to review some algorithmic methods to either parametrize or re-parametrize (rationally and globally) algebraic curves under different optimality criteria. More precisely, we will analyze the computation of rational parametrizations of algebraic curves being optimal in the following sense:

 

  • Properness Optimality: it requires injectivity, i.e. properness.

  • Normal Optimality: it requires surjectivity.

  • Algebraic Optimality: it requires to express the parametrization over the smallest possible field extension of the ground field.

  • Arithmetic Optimality: when Q is a field of parametrization, it requires the smallest possible integer length in the coefficients.

  • Reality questions: If the starting field is F = C, in practical applications one desires to have answers expressed over R. How to decide whether this is possible? Can we achieve the above optimality requirements but now over R?