Symmetric polynomials that detect positivity

Question

Imagine there are numbers $a_1,\ldots,a_n \in \mathbb R$ and you want to know whether they are all positive. You cannot access the numbers themselves, but you can choose any symmetric polynomials you like to evaluate at $(a_1,\ldots,a_n)$. Actually, suppose you don't even get the results of those evaluations, but only the signs.

What is a good family $s_{i,n}$ of symmetric polynomials that allows you to do this?

To restate the problem, we want (for each $n=1,2,...$) a family $s_{1,n},s_{2,n},\ldots,s_{k,n}$ of symmetric polynomials in the variables $x_1,\ldots,x_n$ with the property that these are equivalent:

• $a_j > 0$ for $j=1,\ldots,n$.
• $s_{i,n}(a_1,\ldots,a_n) > 0$ for $i=1,\ldots,k$.

For extra credit, forget about $a_1,\ldots,a_n$ being real, and replace $a_j > 0$ with $\mathfrak{R}(a_j) > 0$.

How do I know this is even possible? It's because a related question has been treated extensively, namely that of determining, given the coefficients of a polynomial, whether all of the roots of the polynomial are in the right (or left or whatever) half-plane. Since the coefficients are (the elementary) symmetric polynomials applied to the roots of the polynomial, results like the Routh–Hurwitz stability criterion, which is usually expressed in terms of the signs of certain polynomials in the coefficients, can be rewritten in terms of symmetric polynomials applied to the roots.

But what if you don't care about going through the coefficients? What if you are happy enough to write down symmetric polynomials in terms of the roots themselves? Then what is the “best” or “most natural” family of symmetric polynomials that will do the job?

Answer 1

The elementary symmetric functions $1=e_0,e_1,\dots,e_n$ will suffice. Clearly $e_j>0$ if each $a_i>0$. Conversely, let $P(x)= \prod_{i=1}^n(x+a_i)$. If some $a_i<0$ and all $e_j>0$ then $0=P(-a_i)=\sum_j e_j (-a_i)^{n-j}>0$, a contradiction.

Incidentally, if we assume only that $a_i\in\mathbb{C}$ we can still determine whether each $a_i$ is a positive real number using $n-1$ additional symmetric functions. See Gantmacher, Matrix Theory, vol. 2, Chapter 15, Section 9.1.