Affine Nash groups over real closed fields

TL;DR

This paper proves that semialgebraically connected affine Nash groups over real closed fields are Nash isogenous to algebraic groups, extending previous results over the real numbers and correcting earlier proof errors.

Contribution

It establishes a general isogeny result for affine Nash groups over any real closed field, broadening the scope of prior work limited to real numbers.

Findings

Proves Nash isogeny to algebraic groups over real closed fields

Extends previous results from real numbers to arbitrary real closed fields

Corrects and generalizes earlier proof with a new, valid approach

Abstract

We prove that a semialgebraically connected affine Nash group over a real closed field R is Nash isogenous to the semialgebraically connected component of the group H(R) of R-points of some algebraic group H defined over R. In the case when R is the field of real numbers this result was claimed in the paper "Groups definable in local fields and pseudofinite fields", Israel J. Math. 85 (1994) by the same two authors, but a mistake in the proof was recently found, and the new proof we obtained has the advantage of being valid over an arbitrary real closed field. We also extend the result to not necessarily connected affine Nash groups over arbitrary real closed fields.