Bounds on symmetric polynomials in power-sum form with bounded coefficients

Question

Let $\boldsymbol{x}=(x_1,\ldots,x_n)$ be a real vector. Define the normalized power-sum symmetric polynomials by $\pi_j(\boldsymbol{x})=\frac 1n(x_1^j+\cdots+x_n^j)$. For a partition $\lambda= (j_1,\ldots,j_t)\vdash k$, define $f_\lambda(\boldsymbol{x}) = \pi_{j_1}(\boldsymbol{x})\cdots\pi_{j_t}(\boldsymbol{x})$.

Obviously, $f_\lambda(\boldsymbol{x})$ is homogeneous of degree $k$ and $\max_{\boldsymbol{x}\in[-1,1]^n} |f_\lambda(\boldsymbol{x})|=1$.

For real constants $\boldsymbol{a}=\{a_\lambda\}$, not necessarily positive, define $$F_k(\boldsymbol{x},\boldsymbol{a})=\sum_{\lambda\,\vdash\, k} a_\lambda f_\lambda(\boldsymbol{x}).$$ The quantity I am interested in is $$ c_{n,k} = \min_{\max|a_\lambda|=1} \max_{\boldsymbol{x}\in[-1,1]^n} |F_k(\boldsymbol{x},\boldsymbol{a})|. $$ I'm mostly interested in $n\to\infty$ with bounded $k$, and a good asymptotic lower bound would probably be just as useful (or useless) in the application in graph enumeration.

For example, when $k=2$ we have $c_{n,2}=1$ for even $n$ and asymptotic to 1 for odd $n$. The optimum is achieved by $\pi_2(\boldsymbol{x}) - \pi_1(\boldsymbol{x})^2$ when $\boldsymbol{x}$ contains an equal number of $+1$ and $-1$.

$c_{n,3}$ might be asymptotically about 0.16, but I haven't proved it.

What's a simply way to prove $\liminf_{n\to\infty} c_{n,k}>0$ for fixed $k$?

Answer 1

This is a (boring) answer to the final question.

Fix $\boldsymbol{y}=(y_1,\ldots,y_k)\in [-1,1]^k$ and choose $x_1,\ldots,x_n$ so that for all $i$ either $\lfloor n/k\rfloor$ or $\lceil n/k\rceil$ of them are equal to $y_i$. Then $|\pi_j(\boldsymbol{x})-\pi_j(\boldsymbol{y})|<k/n$ for $j=1,\ldots,k$. Thus, since we consider the polynomials in $k$ variables on the uniformly bounded set and with uniformly bounded coefficients, for large $n$ the values $F_k(\boldsymbol{x},\boldsymbol{a})$ and $F_k(\boldsymbol{y},\boldsymbol{a})$ become uniformly arbitrarily close. But when $\boldsymbol{y}$ runs over $[-1,1]^k$, the vector $F(\boldsymbol{y})=(\pi_1(\boldsymbol{y}),\pi_2(\boldsymbol{y}),\ldots,\pi_k(\boldsymbol{y}))$ runs over some compact set $C\subset \mathbb{R}^k$ with non-empty interior (the interior is non-empty by the inverse function theorem, since the Jacobian of $F$ is non-zero when all $y_i$'s are distinct). The maximum of absolute value on $C$ is a norm on the space of polynomials of degree at most $k$ in $k$ variables. This space is finite dimensional, thus all norms are equivalent, in particular it is equivalent to the "maximal absolute value of coefficients" norm. Thus the result.