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

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 ℂⁿ 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.

The author was partially supported by the European OpenMath Thematic Network. He is grateful to the referees for their comments, and to Drs Bradford, Corless, Jeffrey and Watt for many useful discussions.