I have a sequence polynomials $P_n(x_1,...,x_n)$ each with as many variable than its index. It turns out that each one has a symmetric sum identically zero. I would be interested to know if such polynomials (with zero symmetric sum) are known families of polynomials.
For me the symmetric sum is not the standard one but this one, for a function $f(x_1,...,x_n)$ of n variables:
$\sum_{\sigma \in S_n} \epsilon(\sigma) \cdot f(x_{\sigma(1)}, x_{\sigma(2)}, \dots, x_{\sigma(n)})$
Where $S_n$ is the permutation group on n symbols and $\epsilon(\sigma) $ is the signature of the permutation $\sigma$
Thanks
I do not know if these have are well-studied polynomials, but I can at least give a complete characterization of them.
A general polynomial in $n$ variables can be written as $$f(x)=\sum_{\alpha} c_\alpha x^\alpha$$ where $\alpha$ ranges over the set of multi-indices $(a_1,a_2,\dots,a_n)$ such that each $a_i$ is a nonnegative integer, and $c_\alpha$ are constants, only finitely many of which are nonzero. The notation $x^\alpha$ refers to the monomial $x_1^{a_1}x_2^{a_2}\dots x_n^{a_n}$.
Let $\def\S{\mathcal S}\S[f]$ denote the symmetric sum of $f$. First, realize that if $\alpha=(a_1,\dots,a_n)$ has any repeated indices, say $a_i=a_j$, then $\S[x^\alpha]=0$. This is because for any permutation $\sigma$, then letting $\sigma(x)$ denote $x_{\sigma_1}x_{\sigma_2}\dots x_{\sigma_n}$, then $\sigma(x^\alpha)$ and $(\sigma\cdot (i\;j))(x^\alpha)$ are the same monomial but with opposite signs in the sum for $\S[x^\alpha]$.
Already, this gives a large class of polynomials whose symmetric sum is zero. However, there are more. Indeed, for each multi-index $\lambda$ whose entries are distinct and strictly decreasing ($\lambda_1>\lambda_2>\dots>\lambda_n$), we can consider the $n!$ monomials whose multi-indices are permutations of $\lambda$: $$ f_\lambda(x)=\sum_{\sigma\in S_n} c_\sigma x^{\sigma(\lambda)} $$ You can then show that $$ \S[f_\lambda]=\left(\sum_{\sigma\text{ even}}c_\sigma-\sum_{\sigma\text{ odd}}c_\sigma\right)\left(\sum_\sigma x^{\sigma(\lambda)}\right) $$ This is zero if and only if the sum of the even coefficients equals the sum of the off ones.
Combining these two observations, we see that a typical polynomial with symmetric sum zero looks something like $$ \underbrace{x_1x_3-x_1x_2^4x_3^4+7x_2^3+8}_{\text{monomials with repeated powers}} + \underbrace{4x_1x_2^4 + 5x_2x_3^4 - 12x_3x_1^2+x_2x_1^4+4x_3x_2^4-2x_1x_3^4}_{\text{groups of permuted monomials w/ distinct powers s.t. sum(even) = sum(odd)}} $$