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Cubics

  • C. Herbert Clemens
Part of the The University Series in Mathematics book series (USMA)

Abstract

Let us now take up the study of the solution set E⊆ℂℙ2 of the equation
$$F\left( {x,\,y,\,z,} \right) = 0,$$
(2.1)
where F is a homogeneous polynomial of degree 3. Again we will assume that the partial derivatives
$$\frac{\partial F}{\partial x},\frac{\partial F}{\partial y},\frac{\partial F}{\partial z}$$
(2.2)
do not all vanish simultaneously. Unlike the degree-2 case, every cubic polynomial with real coefficients has at least one real root, so if the coefficients of F are real, then the solution set to (2.1) in ℝℙ2will always be a smooth curve. The question whether there exists a solution in ℚℙ2if the coefficients of (2.1) are rational is extremely difficult, and there is as yet no known procedure for deciding in general.

Keywords

Inflection Point Meromorphic Function Stereographic Projection Blow Down Cross Ratio 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Copyright information

© Plenum Press, New York 1980

Authors and Affiliations

  • C. Herbert Clemens
    • 1
  1. 1.University of UtahSalt Lake CityUSA

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