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Let $n$ and $p$ be two positive integers. Consider the function $$\max_{n,p}:\{0,\dots,n\}^p\to\{0,\dots,n\}$$ that computes the maximum of a $p$-tuple of integers in the range $\{0,\dots,n\}$. Are there explicit expressions for symmetric polynomials $P_{n,p}\in\mathbb{Q}[\sigma_1,\dots,\sigma_p]$ such that $P_{n,p}(\sigma_1,\dots,\sigma_p)$ interpolates $\displaystyle\max_{n,p}$? Here the $\sigma_i$ are the elementary symmetric polynomials.


The case $p=2$ can be done by hand : $P_{n,2}(\sigma_1,\sigma_2)$ can be described by the formula $$\sum_{s=0}^{2n} \prod_{\substack{a=0\\a\neq s}} \frac{\sigma_1-a}{s-a} \cdot\left( \sum_{i=\max\{s-n,0\}}^{\lfloor s/2\rfloor} (s-i) \prod_{\substack{j=\max\{s-i,0\}\\j\neq i}}^{\lfloor s/2\rfloor} \frac{\sigma_2-j(s-j)}{i(s-i)-j(s-j)} \right)$$ Which you obtain by interpolating the maximum functions along the ``antidiagonals'' $x+y=s$, $0\leq x,y\leq n$.


As far as I can tell, the result from Interpolation for Symmetric Functions is inapplicable here.

I'm interested in this question to study networks. The special case where $n=p=2^N$ is of particular interest to me. Furthermore, I want to allow more invariant polynomials, specifically those that are invariant under a 2-Sylow group of $\mathfrak{S}_n$.

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The following would be proof of existence of such polynomial rather then actual construction (and it's a generalization of your approach).

Define $S_i$ as a set of all possible values $\sigma_i(a)$ for $a\in\{0,1,\ldots, n\}^p$ and define $A=\{0,1\ldots, n\}^p$ Consider any $p$-tuple $a=(a_1, a_2, \ldots, a_p)\in A$ and define a polynomial $L_a\in\mathbb{Q}[\sigma_1,\sigma_2,\ldots,\sigma_p]$ as follows $$ L_a=\prod_{i=1}^{p}\prod_{t_i\in S_i\backslash\{s_i\}}\frac{\sigma_i-t_i}{s_i-t_i}, $$ where $s_i=\sigma_i(a_1, a_2, \ldots, a_p)$ ($L_a$ is well-defined due to Vieta's theorem). The main property of $L_a$ is that $L_a(a)=1$ and $L_a(b)=0$ for any $p$-tuple $b\in A\backslash \{a\}$.

Now, we can define $$ P_{n,p}=\sum_{a\in A}\max\{a_1, a_2, \ldots, a_p\}\cdot L_a. $$ Clearly, $P_{n,p}\in \mathbb{Q}[\sigma_1,\sigma_2,\ldots,\sigma_p]$ and $P_{n,p}(a)=\max\{a_1, a_2, \ldots, a_p\}$ for any $a\in A$.

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  • $\begingroup$ Thank you for your answer. Indeed, that works :) and generalizes my own approach. Ideally I would like something "explicit" (the aim is to plug in special random variables and to compute the expectation). But there might be no ``better'' formula. $\endgroup$ – Olivier Bégassat Apr 30 '19 at 14:42
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    $\begingroup$ The question of existence needs no proof : just take any multivariate polynomial that interpolates the max function, and average it over the symmetric group. The resulting polynomial is symmetrical (and thus a polynomial in the $\sigma_i$) interpolates the max function; $\endgroup$ – Olivier Bégassat Apr 30 '19 at 14:46
  • $\begingroup$ Yes, indeed, existence is trivial. $\endgroup$ – richrow Apr 30 '19 at 16:24

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