The real part of the decomposition of a polynomial and its determinacy (Q1356310)

It is known that the set of topological equivalence classes of real polynomials in \(n\) variables of degree not exceeding \(k\) is a finite set [\textit{T. Fukuda}, Inst. Haut. Étud. Sci., Publ. Math. 46, 87-106 (1976; Zbl 0341.57019)]. The authors prove that for \(n \geq 2\) the set of those classes for all \(k\) is infinite. This statement is an immediate consequence of a simple criterion that describes when homogeneous polynomial-germs defined on \(({\mathbb{R}}^2, 0)\) are \(C^0\)-finitely determined.