We say that a function
![f:\mathbb{R}^k \rightarrow \mathbb{R}](/media/m/7/f/f/7ff22cb9af26ad1e932436f736e4b1fd.png)
is a metapolynomial if, for some positive integer
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
and
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
, it can be represented in the form
![f(x_1,\cdots , x_k )=\max_{i=1,\cdots , m} \min_{j=1,\cdots , n}P_{i,j}(x_1,\cdots , x_k),](/media/m/0/d/9/0d95b3c778f311e5f7ddc3e95e124152.png)
where
![P_{i,j}](/media/m/f/e/4/fe4ddaf8ff941c6072ee8f6c1cec8b8f.png)
are multivariate polynomials. Prove that the product of two metapolynomials is also a metapolynomial.
%V0
We say that a function $f:\mathbb{R}^k \rightarrow \mathbb{R}$ is a metapolynomial if, for some positive integer $m$ and $n$, it can be represented in the form
$$f(x_1,\cdots , x_k )=\max_{i=1,\cdots , m} \min_{j=1,\cdots , n}P_{i,j}(x_1,\cdots , x_k),$$
where $P_{i,j}$ are multivariate polynomials. Prove that the product of two metapolynomials is also a metapolynomial.