27 research outputs found
The real field with an irrational power function and a dense multiplicative subgroup
This paper provides a first example of a model theoretically well behaved
structure consisting of a proper o-minimal expansion of the real field and a
dense multiplicative subgroup of finite rank. Under certain Schanuel
conditions, a quantifier elimination result will be shown for the real field
with an irrational power function and a dense multiplicative subgroup of finite
rank whose elements are algebraic over the field generated by the irrational
power. Moreover, every open set definable in this structure is already
definable in the reduct given by just the real field and the irrational power
function
Extension of the two-variable Pierce-Birkhoff conjecture to generalized polynomials
Let R denote the reals, and let h: R^n --> R be a continuous,
piecewise-polynomial function. The Pierce-Birkhoff conjecture (1956) is that
any such h is representable in the form sup_i inf_j f_{ij}, for some finite
collection of polynomials f_{ij} in R[x_1,...,x_n]. (A simple example is h(x_1)
= |x_1| = sup{x_1, -x_1}.) In 1984, L. Mahe and, independently, G. Efroymson,
proved this for n 2. In this paper we prove an
analogous result for "generalized polynomials" (also known as signomials),
i.e., where the exponents are allowed to be arbitrary real numbers, and not
just natural numbers; in this version, we restrict to the positive orthant,
where each x_i > 0. As before, our methods work only for n < 3.Comment: 16 pages, 4 figure