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The Geometry of ℂn is Important for the Algebra of Elementary Functions

  • James H. Davenport
Conference paper

Abstract

On the one hand, we all “know” tha\(\sqrt {{z^2}} = z\), but on the other hand we know that this is false when z = −1. We all know that ln e x = x, and we all know that this is false when x = 2πi. How do we imbue a computer algebra system with this sort of “knowledge”? Why is it that \(\sqrt x \sqrt y = \sqrt {xy} \) is false in general y =), but \(\sqrt {1 - z} \sqrt {1 + z} = \sqrt {1 - {z^2}} \) is true everywhere? The root cause of this, of course, is that functions such as \(\sqrt {} \) and log are intrinsically multi-valued from their algebraic definition.

It is the contention of this paper that, only by considering the geometry of ℂ (or ℂ n if there are n variables) induced by the various branch cuts can we hope to answer these questions even semi-algorithmically (i.e. in a yes/no/fail way). This poses questions for geometry, and calls out for a more efficient formulation of cylindrical algebraic decomposition, as well as for cylindrical non-algebraic decomposition. It is an open question as to how far this problem can be rendered fully automatic.

Keywords

Riemann Surface Elementary Function Multivalued Function Computer Algebra System Correct Equation 
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.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • James H. Davenport
    • 1
  1. 1.Department of Computer ScienceUniversity of BathBathEngland

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