A remark about homogeneous polynomial maps (Q701322)

This article deals with homogeneous polynomial maps \(F: \mathbb{R}^n\to\mathbb{R}^n\) of degree \(p\). The authors prove the following results. Theorem 1. If \(p\geq 3\), \(n\geq 2\) or \(p=2\), \(n\geq 4\) then there exists a map \(F: \mathbb{R}^n\to\mathbb{R}^n\) of degree \(p\) which is surjective and not proper; if \(p=1\), or \(n= 1\), or \(p=2\), \(n= 2,3\) then every map \(F: \mathbb{R}^n\to \mathbb{R}^n\) of degree \(p\) satisfies the property \(\{x\in \mathbb{R}^n: F(x)= 0\}\neq\{0\}\) implies \(\{F(x): x\in\mathbb{R}^n\}\neq \mathbb{R}^n\). Theorem 2. If \(p\geq 4\), \(n\geq 2\) or \(p=3\), \(n\geq 3\) then there exists a surjective map \(F: \mathbb{R}^n\to \mathbb{R}^n\) of degree \(p\) for which any right inverse map is unbounded; if \(p= 1\), or \(n=1\), or \(p=2\), \(n= 2,3\), or \(p=3\), \(n=2\) then any surjective map \(F: \mathbb{R}^n\to\mathbb{R}^n\) of degree \(p\) admits a bounded right inverse.