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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.

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.

Let $n \geq 3$ 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.

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.

Let $n \geq 3$ 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 $f(x)$?

Let $n \geq 3$ 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.