Algebraic approximation preserving dimension

TL;DR

This paper proves that semialgebraic and semianalytic sets of positive codimension can be locally approximated to any order by algebraic sets of the same dimension, extending algebraic approximation results.

Contribution

It establishes that algebraic approximation preserving dimension applies to semialgebraic and semianalytic sets, broadening previous understanding.

Findings

Semialgebraic sets of positive codimension can be approximated by algebraic sets.

Algebraic approximation preserving dimension applies to semianalytic sets.

Approximation can be achieved to any order.

Abstract

We prove that each semialgebraic subset of $\R^n$ of positive codimension can be locally approximated of any order by means of an algebraic set of the same dimension. As a consequence of previous results, algebraic approximation preserving dimension holds also for semianalytic sets.