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1 results

  • 2 - Basic Real Algebraic Geometry in R2

  • Adam Sheffer, Bernard M. Baruch College, City University of New York
  • Book:
    Polynomial Methods and Incidence Theory
    Published online:
    17 March 2022
    Print publication:
    24 March 2022, pp 25-34

  • Summary

    After seeing some basics of incidence theory, we wish to discuss how polynomial methods are used to study incidences. For that, we first need a basic introduction to algebraic geometry over the reals. In this chapter we focus mainly on the plane, postponing the treatment of higher dimensions to Chapter 4. This allows us to discuss several planar results in Chapter 3, before dealing with more involved algebraic geometry.

    We begin the chapter by introducing varieties and their properties. We then focus on curves in the plane and their properties: degree, irreducible components, connected components, intersections, and more. We conclude the chapter with a polynomial proof of Pascal’s theorem, due to Plücker.