2012 IMO Shortlist A7

We say that a function $f: \mathbb{R}^{k} \rightarrow \mathbb{R}$ is a metapolynomial if, for some positive integers $m$ and $n$, it can be represented in the form $$f\left(x_{1}, \ldots, x_{k}\right)=\max _{i=1, \ldots, m} \min _{j=1, \ldots, n} P_{i, j}\left(x_{1}, \ldots, x_{k}\right)$$ where $P_{i, j}$ are multivariate polynomials. Prove that the product of two metapolynomials is also a metapolynomial.

我们说函数 $f: \mathbb{R}^{k} \rightarrow \mathbb{R}$ 是元多项式，如果对于某些正整数 $m$ 和 $n$，它可以表示为 $$f\left(x_{1}, \ldots, x_{k}\right)=\max _{i=1, \ldots, m} \min _{j=1, \dots, n} 的形式P_{i, j}\left(x_{1}, \ldots, x_{k}\right)$$ 其中 $P_{i, j}$ 是多元多项式。证明两个元多项式的乘积也是元多项式。