Optimal Parametrizations of Algebraic Curves
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TALK
Prof. Rafael Sendra (University Madrid), 22 April 2010, 2:30 p.m., RISC seminar room
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:
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Properness Optimality: it requires injectivity, i.e. properness.
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Normal Optimality: it requires surjectivity.
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Algebraic Optimality: it requires to express the parametrization over the smallest possible field extension of the ground field.
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Arithmetic Optimality: when Q is a field of parametrization, it requires the smallest possible integer length in the coefficients.
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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?