Let $f\in\mathbb{R}[X_1,\ldots,X_n]$ be symmetric (or more generally invariant by a compact group). Then, there exists $n$ symmetric polynomials (more generally, $m$ $G$-invariant polynomial) - for example, the elementary symmetric polynomials $\sigma_1,\ldots,\sigma_n$ as mentioned by Bazin above, such that every symmetric polynomial can be (uniquely in the case of $G=S_n$) written as $f=g(\sigma_1,\ldots,\sigma_n)$. Now, the fact that $f$ is non-negative amounts to the same as saying that $g$ is non-negative on the image of the map $x\mapsto (\sigma_1,\ldots, \sigma_n)$. Procesi and Schwarz 1 -or in fact already earlier Procesi 2 in the case of symmetric polynomials, make the link to Hilbert's 17th problem: The image can be described as a basic semi-algebraic set, given by explicit inequalities, say $h_1\geq 0,\ldots, h_k\geq 0$. So one gets directly (see for example Theorem 2.1 in Procesi's paper) that $g=\sum_{1\leq i_1<i_2<\ldots<i_t\leq k} s_{i_1\cdots i_t} h_{i_1}h_{i_2}\cdots h_{i_t}$, where each $s_{i_1\cdots i_t}$ is a sum of squares of elementary symmetric functions. This is maybe the most general statement one can hope for, in the case of symmetric polynomials - and generally. Let me also add, that one can use in general representation theory to get a good handle over the cones of symmetric sums of squares of given degree - in particular in the case of the symmetric group and more generally finite reflection groups, where one has a good control, since the representation in terms of the generators is unique. I also advertise my two papers on this topic here [3,4] and happily answer any question

Let $f\in\mathbb{R}[X_1,\ldots,X_n]$ be symmetric (or more generally invariant by a compact group). Then, there exists $n$ symmetric polynomials (more generally, $m$ $G$-invariant polynomial) - for example, the elementary symmetric polynomials $\sigma_1,\ldots,\sigma_n$ as mentioned by Bazin above, such that every symmetric polynomial can be (uniquely in the case of $G=S_n$) written as $f=g(\sigma_1,\ldots,\sigma_n)$. Now, the fact that $f$ is non-negative amounts to the same as saying that $g$ is non-negative on the image of the map $x\mapsto (\sigma_1,\ldots, \sigma_n)$. Procesi and Schwarz 1 -or in fact already earlier Procesi 2 in the case of symmetric polynomials, make the link to Hilbert's 17th problem: The image can be described as a basic semi-algebraic set, given by explicit inequalities, say $h_1\geq 0,\ldots, h_k\geq 0$. So one gets directly (see for example Theorem 2.1 in Procesi's paper) that $g=\sum_{1\leq i_1<i_2<\ldots<i_t\leq k} s_{i_1\cdots i_t} h_{i_1}h_{i_2}\cdots h_{i_t}$, where each $s_{i_1\cdots i_t}$ is a sum of squares of elementary symmetric functions. This is maybe the most general statement one can hope for, in the case of symmetric polynomials - and generally. Let me also add, that one can use general representation theory of linear groups to get a good handle over the cones of symmetric ($G$-invariant) sums of squares of a given degree - in particular in the case of the symmetric group and more generally finite reflection groups, where the representation in terms of the generators is unique. Let me shamelessly also advertise the papers on invariant sums of squares decompositions - in the case of the symmetric group 3 and in the (slightly) more general case 4.

Let $f\in\mathbb{R}[X_1,\ldots,X_n]$ be symmetric (or more generally invariant by a compact group). Then, there exists $n$ symmetric polynomials (more generally, $m$ $G$-invariant polynomial) - for example, the elementary symmetric polynomials $\sigma_1,\ldots,\sigma_n$ as mentioned by Bazin above, such that every symmetric polynomial can be (uniquely in the case of $G=S_n$) be written as $f=g(\sigma_1,\ldots,\sigma_n)$. Now, the fact that $f$ is non-negative amounts to the same as saying that $g$ is non-negative on the image of the map $x\mapstp (\sigma_1,\ldots, \sigma_n)$. Procesi and Schwarz 1 -or in fact already earlier Procesi 2 in the case of symmetric polynomials, make the link to Hilbert's 17th problem: The image can be described as a basic semi-algebraic set, given by explicit inequalities, say $h_1\geq 0,\ldots, h_k\geq 0$. So one gets directly (see for example Theorem 2.1 in Procesi's paper) that $g=\sum_{1\lleq i_1<i_2<\ldots<i_t\leq k} s_{i_1\cdots i_t} h_{i_1}h_{i_2}\cdotsh_{i_t}$, where each $s_{i_1\cdots i_t}$ is a sum of squares of elementary symmetric functions. This is maybe the most general statement one can hope for, in the case of symmetric polynomials - and generally. Let me also add, that one can use in general representation theory to get a good handle over the cones of symmetric sums of squares of given degree - in particular in the case of the symmetric group and more generally finite reflection groups, where one has a good control, since the representation in terms of the generators is unique. I also advertise my two papers on this topic here [3,4] and happily answer any question

Let $f\in\mathbb{R}[X_1,\ldots,X_n]$ be symmetric (or more generally invariant by a compact group). Then, there exists $n$ symmetric polynomials (more generally, $m$ $G$-invariant polynomial) - for example, the elementary symmetric polynomials $\sigma_1,\ldots,\sigma_n$ as mentioned by Bazin above, such that every symmetric polynomial can be (uniquely in the case of $G=S_n$) written as $f=g(\sigma_1,\ldots,\sigma_n)$. Now, the fact that $f$ is non-negative amounts to the same as saying that $g$ is non-negative on the image of the map $x\mapsto (\sigma_1,\ldots, \sigma_n)$. Procesi and Schwarz 1 -or in fact already earlier Procesi 2 in the case of symmetric polynomials, make the link to Hilbert's 17th problem: The image can be described as a basic semi-algebraic set, given by explicit inequalities, say $h_1\geq 0,\ldots, h_k\geq 0$. So one gets directly (see for example Theorem 2.1 in Procesi's paper) that $g=\sum_{1\leq i_1<i_2<\ldots<i_t\leq k} s_{i_1\cdots i_t} h_{i_1}h_{i_2}\cdots h_{i_t}$, where each $s_{i_1\cdots i_t}$ is a sum of squares of elementary symmetric functions. This is maybe the most general statement one can hope for, in the case of symmetric polynomials - and generally. Let me also add, that one can use in general representation theory to get a good handle over the cones of symmetric sums of squares of given degree - in particular in the case of the symmetric group and more generally finite reflection groups, where one has a good control, since the representation in terms of the generators is unique. I also advertise my two papers on this topic here [3,4] and happily answer any question