Approximation of maps between real algebraic varieties

Abstract A nonsingular real algebraic variety 𝑌 is said to have the approximation property if, for every nonsingular real algebraic variety 𝑋, the following holds: if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>f</m:mi> <m:mo lspace="0.278em" rspace="0.278em">:</m:mo> <m:mrow> <m:mi>X</m:mi> <m:mo stretchy="false">→</m:mo> <m:mi>Y</m:mi> </m:mrow> </m:mrow> </m:math> f\colon X\rightarrow Y is a <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="script">C</m:mi> <m:mi mathvariant="normal">∞</m:mi> </m:msup> </m:math> \mathcal{C}^{\infty} map that is homotopic to a regular map, then 𝑓 can be approximated in the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="script">C</m:mi> <m:mi mathvariant="normal">∞</m:mi> </m:msup> </m:math> \mathcal{C}^{\infty} topology by regular maps. In this paper, we characterize the varieties 𝑌 with the approximation property. We also characterize the varieties 𝑌 with the approximation property combined with a suitable interpolation condition. Some of our results have variants concerning the regular approximation of continuous maps defined on (possibly singular) real algebraic varieties.