Literature about metapolynomials 
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.

This problem was given in the IMO Shortlist 2012 A7, I want to know if there is some paper on this topic metapolynomials. I searched on the web but I found nothing. I found this topic interesting. So a reference would be helpful.
 A: If you put an absolute value around your polynomial, then this sort of max-min comes up in the construction of local height functions. In the terminology of Lang's Fundamentals of Diophantine Geometry (see especially Chapter 10, Theorem 3.5, page 261), the equation displayed at the top of page 262 is
$$
  \lambda = \sup_j \inf_i \lambda_{ij}.
$$
Unsorting the definitions, one has essentially
$$
  \lambda_{ij}(x_1,\ldots,x_n) = \log\bigl| f_{ij}(x_1,\ldots,x_n)\bigr|
$$
for a certain collection of polynomials (or maybe rational functions) $f_{ij}$. So aside from having taken logs, the function $\lambda$ is the IMO max-min with absolute values.
These Weil functions are associated to divisors and do satisfy an addition formula $\lambda_{D_1}+\lambda_{D_2}=\lambda_{D_1+D_2}$, which is related to the IMO question. The IMO question is less general, but more precise because the Weil function $\lambda_D$ associated to a divisor $D$ is only well-defined up to what Lang calls an $M_K$-bounded function. On the other hand, the proof of the addition formula probably comes down to doing the IMO problem (modulo those pesky absolute value signs).
Final Comment: In Lang's setup, the absolute value may be the usual one on $\mathbb R$ or $\mathbb C$, but it could also be a $p$-adic absolute value; and more generally, one really wants a collection of Weil height $\lambda_{D,v}$, one for each absolute value $v$, that have some sort of uniformity as one varies over $v$.
A: The Pierce–Birkhoff conjecture states that any (continuous) piecewise-polynomial function is a so called metapolynomial.
A search in Google for "Pierce-Birkhoff conjecture" gives some articles discussing this conjecture and related conjectures.
