October 4
Hans Schoutens,
CUNY
All your favorite Taylor series wrapped up in a nice little package: the ring of 'catanomials'.
The nice little package is a regular, existentially closed, Henselian local subring of the ring of (formal) power series over your favorite field (R, C, Q?). Moreover, this ring is closed under derivations, anti-derivations, composition, etc. The favorite series (in a single variable, say) include all algebraic functions, all elementary functions, all hypergeometrical functions, all holonomic functions (i.e., solutions of a linear, algebraic ODE), etc.
The way to obtain these is by looking at some non-standard model of the theory of polynomial rings, and then defining its 'catanomials' as the truncations of these functions by only looking at its finite degree terms. In the special case that the non-standard model is an ultrapower of the polynomial ring, the resulting algebra, called the catapower, is just the full power series ring. Whereas the latter may sound less glorious, we can nonetheless do better by taking different non-standard models. Enters the embedded model of PA* of such a model and its standard systems!