On the concavity of a sum of elementary symmetric polynomials

We introduce a new problem on the elementary symmetric polynomials $\sigma_k$, stemming from the constraint equations of some modified gravity theory. For which coefficients is a linear combination of $\sigma_k$ $1/p$-concave, with $0 \leq k \leq p$? We establish connections between the $1/p$-concavity and the real-rootedness of some polynomials built on the coefficients. We conjecture that if the restriction of the linear combination to the positive diagonal is a real-rooted polynomial, then the linear combination is $1/p$-concave. Using the theory of hyperbolic polynomials, we show that this would be implied by a short algebraic statement: if the polynomials $P$ and $Q$ of degree $n$ are real-rooted, then $\sum_{k=0}^n P^{(k)}Q^{(n-k)}$ is real-rooted as well. This is not proven yet. We conjecture more generally that the global $1/p$-concavity is equivalent to the $1/p$-concavity on the positive diagonal. We prove all our guessings for $p=2$. The way is open for further developments.