Example where the polarization of a homogeneous function fails to be multilinear.

Question

Let $V$ and $W$ be $K$-modules, where $K$ is a commutative ring. Then given a function $Q: V \to W$ which is homogeneous of degree $p$ (i.e. $Q(tv)=t^pQ(v)$ for all $t \in K$) we can write down its polarization $$m: V^p \to W$$ (as shown in the nLab article or this MSE question) such that $m \circ \Delta = p! Q$ where $\Delta: V \to V^p$ is the diagonal map. In the nLab article they state that

So defined, $m$ is manifestly symmetric, but it might not be multilinear just because $Q$ is homogeneous (except for $p=1$); instead, we define a homogeneous polynomial of degree $p$ from $V$ to $W$ be such a homogeneous $Q$ such that $m$ is multilinear.

I'm curious if there are some simple examples of homogeneous functions such that their polarization fails to be multilinear.

Answer 1

Take $Q:\mathbb R^2\to \mathbb R$, $Q([x,y])=\max(x^2,y^2)$, which is homogeneous of degree 2, whose polarization $m([x_1,y_1],[x_2,y_2])= \max((x_1+x_2)^2,(y_1+y_2)^2)-\max(x_1^2,y_1^2)-\max(x_2^2,y_2^2)$ is not bilinear. (I am not sure why they exclude p=1, as homogeneity of degree 1 does not imply additivity.)