Bruce gegen Hilbert '88

In Hilbert's celebrated 1888 paper, he gave a an enigmatically
nonconstructive proof that a positive ternary quartic was the sum
of three squares.  Bruce Reznick's influential oeuvre on positivity
and sums of squares took a different, decidedly constructive, 
approach to positivity.

I will relate a story of my interaction with Bruce and Hilbert's
theorem that began with a mind-blowing computation of Powers and
Reznick.  This led to a most enjoyable series of lunches during
a conference at Rennes in 2001, and to an entirely new proof and 
explicit form of Hilbert's Theorem, that a smooth positive ternary
quartic has essentially eight representations as a sum of squares.