Question 39
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.
$$
f\left(x_1, \ldots, x_k\right)=\max_{i=1, \ldots, m} \min_{j=1, \ldots, n} P_{i, j}\left(x_1, \ldots, x_k\right)
$$
where $P_{i, j}$ are multivariate polynomials. Prove that the product of two metapolynomials is also a metapolynomial.