On Zero Problems in Real Computation
(Or: Why talk about nothing?)

Chee Yap

ABSTRACT:

The zero problem is that of deciding
whether a given numerical expression represents the value zero.
Such problems come in many guises,
from correct curve tracing to robust geometric computation.
Each zero problem is defined by the underlying set of expressions.
The decidability of these zero problems is
equivalent (in a certain precise sense) to being
being able to compute exact geometry
within the corresponding real subfield.
We outline a theory of real computation which 
is suitable for treating such zero problems.