Connected polynomials and continuity (Q1754585)

Given a normed space \(E\), a function \(P: E\to \mathbb{K}\), \( \mathbb{K}= \mathbb{R}\) or \( \mathbb{C}\), is said to be an \(n\)-homogeneous polynomial if there is an \(n\)-linear mapping \(A: E\times\cdots\times E\to \mathbb{K}\) such that \(P(x)= A(x,\dots,x)\) for all \(x\) in \(E\). A polynomial is a sum of homogeneous polynomials. In [\textit{J. L. Gámez-Merino} et al., Linear Algebra Appl. 436, No. 1, 237--242 (2012; Zbl 1242.46057)], it has been shown that, if \(E\) is a real normed space, then an \(n\)-homogeneous polynomial \(P\) on \(E\) is continuous if and only if it maps compact subsets of \(E\) to compact subsets of \( \mathbb{K}\). In this paper, the authors show that, if \(E\) is a complex normed space, then a polynomial \(P\) on \(E\) is continuous if and only if \(P\) maps connected subsets of \(E\) to connected subsets of \(\mathbb{C}\). They also prove the corresponding result for polynomials of degree at most \(2\) on real normed spaces.