Resultant of $f(x)$ and $f(-x)$

Question

Let $n \geq 1$ be an integer, and let $f(x) = \prod\limits_{k = 1}^n(x - \alpha_k)$ be a monic polynomial in $\mathbb Z[x]$.

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 if $f$ is irreducible, then $f(-\alpha_1), \ldots, f(-\alpha_n)$ 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.

Answer 1

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 $$ \left| \begin{array}{cc} a_2 & a_3\\ 1 & a_1\end{array} \right| =a_2a_1-a_3. $$

Answer 2

See the discussion above.

Also, if $f(x) = x^n + \dotsb + a_{n-1}x + a_n$, then

• for $n = 2$, $\operatorname{Res}(f(x), f(-x)) = 2^2a_2a_1^2$

• for $n = 3$, $\operatorname{Res}(f(x), f(-x)) = 2^3a_3(a_3 - a_1a_2)^2$

• for $n = 4$, $\operatorname{Res}(f(x), f(-x)) = 2^4a_4(a_4a_1^2 - a_1a_2a_3 + a_3^2)^2$.