Algebra, Geometry and Software Systems pp 207-224 | Cite as

# The Geometry of ℂ^{n} 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* = 2*πi*. 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## Preview

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