Stratifications of tangent cones in real closed (valued) fields

TL;DR

This paper generalizes tangent cones to subsets of real closed fields, studies how non-archimedean stratifications influence these cones, and shows that such stratifications induce Whitney stratifications in the archimedean setting.

Contribution

It introduces tangent cones in real closed fields and demonstrates how t-stratifications induce Whitney stratifications on tangent cones of semi-algebraic sets.

Findings

Tangent cones are generalized to real closed fields.

T-stratifications induce stratifications on tangent cones.

Archimedean t-stratifications induce Whitney stratifications.

Abstract

We introduce tangent cones of subsets of cartesian powers of a real closed field, generalising the notion of the classical tangent cones of subsets of Euclidean space. We then study the impact of non-archimedean stratifications (t-stratifications) on these tangent cones. Our main result is that a t-stratification induces stratifications of the same nature on the tangent cones of a definable set. As a consequence, we show that the archimedean counterpart of a t-stratification induces Whitney stratifications on the tangent cones of a semi-algebraic set. The latter statement is achieved by working with the natural valuative structure of non-standard models of the real field.