FOM: Measures of semi algebraic sets

[Lou van den Dries]

This concerns Harvey Friedman's question in what sense measure is
entirely well behaved for semi algebraic sets. I considered this
question several years ago, and was led to conjecture that it
behaves very well with respect to variation of parameters, more
specifically, if Y is a bounded semialgebraic subset of R^{m+n}, some experimentation and plausible arguments (which I could not quite make to
stick) suggested that the volume of Y_x := {y in R^n: (x,y) in Y}
as x ranges over the parameter space R^m is a function of the form

P(f_1(x),...,f_k(x), log f_1(x),...,log f_k(x))

where P is a polynomial, and f_1,...,f_k are functions on R^m that are
definable in the o-minimal structure R_an, that is, the field of reals
expanded by the "restricted analytic functions". 
(The actual conjecture was about integrals of bounded functions
depending on parameters, the functions being definable in R_an.)
 Somewhat later the analytic geometers Lion and Rolin came out with
a paper proving geometrically some results from 
my article with Macintyre and Marker in Ann. Math. 140 (1994)
(which we had obtained model-theoretically), and Lion and Rolin actually
obtained these results in somewhat sharper form. I suspected that
this extra strength was exactly what was needed to prove the
conjecture above, and when I met Lion and Rolin in March I asked
if they could prove it by their methods, and they did:

Lion and Rolin, Volumes des sous-ensembles sous-analytiques, April '97
(you can ask lion at u-bourgogne.fr).

Note that in particular the volume function is definable in the
*o-minimal* structure (R_an, exp). Thus as a special case,

if Y and Z are bounded semialgebraic subsets of R^{1+n}, then the
set of real numbers x such that Y_x and Z_x have the same volume
(as subsets of R^n) is a finite union of intervals and points.

(The restriction to bounded sets and bounded functions in the above
is not essential, it only makes it easier to state the results.)


If one looks at individual semialgebraic sets defined by polynomials
with rational coefficients, one is soon in big trouble:
the disc of radius 1 has area \pi, the area under the curve
y=1/x and above the x-axis from x=1 to x=r is log r, other
integrations bring in arctangents and arcsines of rational numbers,
and taking cartesian products multiplies volumes (and taking
disjoint unions adds them). Soon is even beyond the realm of
elementary functions, since elliptic integrals and more
generally values of abelian integrals with rational integration
bounds have to be faced. To decide equality of numbers built up
polynomially from such quantities seems only a realistic problem
once there has been earthshaking progress in transcendental number
theory. In any case, faced with such intractable (at the moment)
decision problems, the way to make it into a more fruitful question
is to vary the parameters and study the dependence on parameters.
(In fact, also in transcendental number theory one ultimately
arrives at results by doing things of this kind.)

-Lou van den Dries-