Hilbert's Tenth Problem and the Genus Zero Locus

		      J. Maurice Rojas

		City University of Hong Kong 

	While Hilbert's Tenth Problem has been settled negatively, 
the exact frontier to undecidability is still unknown.  So we will 
consider the following question: 

	What is the least d such that one can NOT algorithmically 
	determine whether an ARBITRARY variety of dimension d has an 
	integral point? 

Until recently, it was only known that 1<d<12. 
	We show evidence that d=2.  In particular, we show that 
if d>2, then (under an additional technical assumption) there must 
exist algebraic plane curves with uncomputably large integral 
points. The vast effort number theory has devoted to deriving 
height bounds on many (but not all) curves can then 
be taken as controversial evidence that d=2. 
	The proof is based on some recent fast algorithms 
for algebraic geometry over the complex numbers.  We begin 
by briefly reviewing these results, and then outlining the 
proof of our main result.  Time permitting, we will also 
present a result (depending on the truth of the Generalized 
Riemann Hypothesis) on the complexity of detecting rational points 
on zero-dimensional varieties.