resultant of two co-prime polynomials over $\mathbb{Z}[x]$ , is there anything special about this resultant?

Question

let $f(x)=x^3-1$ and $g(x)=a_0+a_1x+a_2x^2$, where $f(x),g(x) \in \mathbb{Z}[x]$, and $\textrm{GCD}(f(x),g(x))=1$. From all this, is it possible to infer anything about the resultant $\textrm{Res}(f(x),g(x))$? I know it has to be non-zero and also $a_0+a_1+a_2 =1$ (i proved this bit using other properties).but I was wondering if there was any other property the resultant had. I ask because i eventually want to solve the diophantine equation $\textrm{Res}(f(x),g(x))= 1$, this equation is not easy to solve as n increases so i was hopping there were other properties of the resultant i could exploit to make the equation simpler

Thanks in advance

Answer 1

let $f(x)=x^3-1, g(x)=a_0+a_1x+a_2x^2$, $f(x),g(x) \in \mathbb{Z}[x]$, and $GCD(f(x),g(x))=1$. From all this, is it possible to infer anything about the resultant $Res(f(x),g(x))$? I know it has to be non-zero

Indeed, $\,\operatorname{Res}(f(x),g(x)) \ne 0\,$ follows from $\,\gcd(f(x),g(x))=1\,$, because the resultant is $\,0\,$ iff the two polynomials have at least one common root.

and also $\,a_0+a_1+a_2 \neq 0\,$ (otherwise $\,\operatorname{Res}(f(x),g(x))=0\,$)

The condition also follows from $\,\gcd(f(x),g(x))=1\,$, because $\,a_0+a_1+a_2 = 0$ $\iff \,g(1) = 0\,$, and in that case $\,(x-1) \mid \gcd(f(x),g(x))\,$ since $\,1\,$ is also a root of $\,f(x) = x^3 -1\,$.

By the same $\,\gcd\,$ condition, the complex cube roots of unity, which are the other two roots of $\,f(x)\,$, must not also be roots of $\,g(x)\,$, which is equivalent to $\,a_0, a_1, a_2\,$ not being all three equal.

but i was wondering if there was any other property the resultant had. I ask because i eventually want to solve the equation $Res(f(x),g(x))= 1$

It is not clear what other properties you are looking for, or what unknown(s) the equation you mention at the end would be solved for. But the resultant can be calculated explicitly, so you can look at it and assess yourself whether it has the desired properties. This can be done either with the help of a CAS (for example, WA), or directly by hand (see e.g. 1, 2), and the result is:

$$ \begin{align} \operatorname{Res}(f(x),g(x)) &= a_0^3 + a_1^3 + a_2^3 - 3 a_0 a_1 a_2 \\ &= (a_0 + a_1 + a_2) (a_0^2 + a_1^2 + a_2^2 - a_0 a_1 - a_0 a_2 - a_1 a_2) \\ &= \frac{1}{2}(a_0 + a_1 + a_2) \big((a_0-a_1)^2 + (a_0-a_2)^2 + (a_1-a_2)^2\big) \end{align} $$

Answer 2

The resultant is get using the old Sylvester's method. We have $$x^3+0x^2+0x-1=0\\a_2x^2+a_1x+a_0=0$$Then the resultant is an homogenous function, rational, integer, of $3$ degree respect to the coefficients of second polynomial and of $2$ degree respect to those of the first one.

$$\begin{vmatrix}1&0&0&-1&0\\0&1&0&0&-1\\a_2&a_1&a_0&0&0\\0&a_2&a_1&a_0&0\\0&0&a_2&a_1&a_0\end{vmatrix}=0$$ So you have $$a_2^3+a_1^3+a_0^3-3a_2a_1a_0=0$$

I think the other properties you want should be deduced, (if any interesting and useful) from this resultant itself.