Topological bi-$\mathcal{K}$-equivalence of pairs of map germs

Abstract

Let $P^{k}(n,p \times q)$ be the set of all pairs of real polynomial map germs $(f, g) : (\mathbb{R}^{n},0) \rightarrow (\mathbb{R}^{p} \times \mathbb{R}^{q} ,0)$ with degree of $ f_1 , \dots, f_p ,$ $g_1 ,\dots, g_q$ less than or equal to $k \in \N$. The main result of this paper shows that the set of equivalence classes of $P^{k}(n,p \times q)$, with respect to bi-$C^{0}$-$\mathcal{K}$-equivalence, is finite.