Let $n \geq 2$ be an integer, and let $f(x) = \prod\limits\_{k = 1}^n(x - \alpha\_k)$ be a monic irreducible polynomial in $\mathbb Z[x]$, with the property that $f(-\alpha\_k) \neq 0$ for any $k = 1, 2, \ldots, n$.


Is there anything meaningful that we can say about $\operatorname{Res}(f(x), f(-x))$, the resultant of $f(x)$ and $f(-x)$?


To rephrase, what can be said about the value of the product $\prod\limits\_{k = 1}^nf(-\alpha\_k)$? Perhaps, it can be expressed somehow through $n$ and the discriminant of $f(x)$?


One thing that I can note is that $f(-\alpha\_1), \ldots, f(-\alpha\_k)$ are algebraic conjugates, which means that their product is equal to the norm of $f(-\alpha\_1)$. Thus, up to a sign, the product $\prod\limits\_{k = 1}^nf(-\alpha\_k)$ is equal to the constant coefficient of the minimal polynomial of $f(-\alpha\_1)$. But what is this constant coefficient is a mystery.



Resultant of $f(x)$ and $f(-x)$
We have $\mathrm{Res}(f(x),f(-x))=2^n a\_n P(\alpha)^2$, where
$P(\alpha)=\prod\_{1\leq i<j\leq n}(\alpha\_i+\alpha\_j)$. By
e.g. the case $d=2$ of Exercise 7.30 in *Enumerative Combinatorics*,
vol. 2, we have $P(\alpha)=s\_{n-1,n-2,\dots,1}(\alpha)$, where
$s\_{n-1,n-2,\dots,1}$ is a Schur function. By the dual Jacobi-Trudi
identity (Corollary 7.16.2 of the above reference), we get
$$ \mathrm{Res}(f(x),f(-x)) = 2^n a\_n
 \left( \det[a\_{n-2i+j}]\_{i,j=1}^{n-1}\right)^2, $$
where we set $a\_0=1$ and $a\_{-k}=0$ for $k>0$. For instance, when
$n=3$ the determinant is
$$ \left| \begin{array}{cc} a\_2 & a\_3\\ 1 & a\_1\end{array} \right|
 =a\_2a\_1-a\_3. $$


