Suppose that I have a function $f(x_1,x_2,\ldots,x_n)$ that is a multivariate polynomial linear in each argument, i.e. each additive term contains $x_i$ raised to the power 1 or zero, but no more than one. Formally, for the set $\mathcal{B}_n$ of all $n$-bit binary vectors, and vector $\beta\in \mathcal{B}_n$, denote $\beta_i$ as the $i$-th element of $\beta$ (either 0 or 1). Then, $f(x_1,x_2,\ldots,x_n)=\sum_{\beta\in \mathcal{b}_n}c_\beta\prod_{i=1}^nx_i^{\beta_i}$, where $c_\beta$ is a coefficient associated with each $\beta$.
Now suppose also that $f(\cdot)$ is shift-symmetric, that is: $f(x_1,x_2,\ldots,x_n)=f(x_2,x_3,\ldots,x_n,x_1)=\ldots=f(x_n,x_1,\ldots,x_2)$
I am interested in properties of the stationary points of such $f(\cdot)$ for non-negative arguments $x_i\geq 0$. It seems to me (from experimentation) that, if they exist (see comment from @copper.hat) at least one is achieved with equal $x_i$'s: $x_1=x_2=\ldots=x_n$ (at least in the cases I tested). I am wondering if this is true for all functions $f(\cdot)$ defined above, and if so, how to prove that. Also, when it exists, does each $f(\cdot)$ have a unique non-negative stationary point? ($f(\cdot)$ that has two stationary points, one being negation of the other, is pointed out by @copper.hat in the comments) I am also interested in the conditions for $f(\cdot)$ for existence of at least one stationary point (I think that has to do with the signs of the coefficients).
This problem arises in the study of the trace of the inverse of the sum of circulant and diagonal matrices... Reason I ask this is because I can't find any literature on the properties of the shift-symmetric multivariate polynomials. Maybe I am not looking in the right places.
