2
i. If f:
n
m
V, g:
m
p
V, then h:
n
p
V,
where for all x
n
, h(x) = g(f(x)).
ii. If f:
n
m
V, g:
n
p
V, then h:
n
m+p
V,
where for all x
n
, h(x) = (f(x),g(x)).
4. ZERO SELECTION ( ). Let f:
n+m
V. There exists
g:
n
m
V such that if f(x,y) = 0 then f(x,g(x)) = 0.
We use the adjective “special” if, in addition, we have
5. LIMIT (
). Every bounded f:
V has a limit at
infinity.
From model theory, an expansion of the ring of real numbers
is a system (
,<,0,1,+,-,•,V), where V obeys MULTIVARIATE
(
).
From model theory, an o-minimal expansion of the ring of
real numbers is an expansion of the ring of real numbers in
which every first order definable subset of is a finite
union of open intervals and points. Here and - are
allowed as endpoints. Here, and throughout the paper,
"definable" means "definable with parameters by a formula
in the first order predicate calculus with equality".
The above definition is a special case of the more general
definition of o-minimal structure introduced in [PS86],
which we use in section 4.
Specifically, a linearly ordered structure is a system
(D,<,V), where (D,<) is a strictly linear ordering, and V
is a set of constants from D, multivariate relations on D,
and multivariate functions from D into D. An o-minimal
structure is a structure M = (D,<,V), where every M
definable subset of D is a finite union of open intervals
and points, where the endpoints of the intervals lie in D
{- , } and the points lie in D.
We prove the following.