How to evaluate $|a_1|^2+|a_2|^2+|a_3|^2+\dots$ using elementary symmetric polynomials.

Question

Given the values of the elementary symmetric polynomials $$e_1(a_1,a_2,\dots,a_n),\, e_2(a_1,a_2,\dots,a_n),\, \dots,\, e_n(a_1,a_2,\dots,a_n)$$ of $n$ complex numbers $a_i\in\mathbb C,$ I want to find the value of $$|a_1|^2+|a_2|^2+\dots+|a_n|^2.$$ I know this can be easily done when $a_1\in\mathbb R,$ because it simplifies to $$\sum_{i=1}^n a_i^2=\left(\sum_{i=1}^n a_i\right)^2-2\sum_{1\leq i\leq j\leq n}a_ia_j=e_1^2-2e_2,$$ but with complex numbers I can't find such an expression.

Answer 1

$\sum |a_k|^2$ is not a polynomial function in $a_k$, and not every symmetric but non-polynomial function can be expressed in terms of the elementary symmetric polynomials.

Taking for example the case $n=2$ with $e_1, e_2 \in \mathbb R$:

• if the roots are real then $|a_1|^2 + |a_2|^2 = a_1^2 + a_2^2 = (a_1+a_2)^2 - 2a_1a_2 = e_1^2 - 2 e_2$;

• if the roots are complex (which implies $e_2 \gt 0$) then they must be conjugates, so $|a_1|=|a_2|=\sqrt{e_2}$ and $|a_1|^2 + |a_2|^2 = 2 e_2$.

Since the expressions are not the same in the two cases, there exists no polynomial or rational function in $e_1, e_2$ that represents $|a_1|^2+|a_2|^2$, even if conjugates $\bar e_1, \bar e_2$ are also allowed.