On the norm of the complexification of polynomials

We investigate various complexification procedures on the algebraic complexification $\widetilde {E}$ of a real Banach space $E$, focusing on the extension of polynomials. Specifically, we analyze the norm of the complex extension $\widetilde {P}$ of a homogeneous polynomial $P: E \rightarrow \mathbb {R}$, considering different norms on $\widetilde {E}$. To facilitate our study, we introduce the concept of complexification constants. Among our findings, we establish that for Bochnak’s complexification procedure, the space $\ell _1$ has the largest possible complexification constants. This means that $\widetilde {\ell _1}$ is where the norm of polynomials can increase the most. For Hilbert spaces, we prove that the only complexification procedure that preserves the norm of homogeneous polynomials is Bochnak’s complexification procedure. We also show that when employing Taylor’s complexification procedure, all complexification constants are away from 1, provided the space has dimension at least 2. In terms of polynomials, this implies the existence of a polynomial $P$ whose complex extension $\widetilde P$ has a norm significantly different from that of $P$.