O kvaternionech. [II.]
Let R be an o-minimal field and V a proper convex subring with residue field k and standard part (residue) map st: V → k. Let be the expansion of k by the standard parts of the definable relations in R. We investigate the definable sets in and conditions on (R,V) which imply o-minimality of . We also show that if R is ω-saturated and V is the convex hull of ℚ in R, then the sets definable in are exactly the standard parts of the sets definable in (R,V).
One considers representation of a polynomial in several variables as the sum of values of univariate polynomials taken at linear combinations of the variables.
One considers representation of cubic polynomials in several variables as the sum of values of univariate polynomials taken at linear combinations of the variables.