Resultant $(g(x),h(x)) = 0$ iff $\deg(\gcd(g(x),h(x)) )> 0$

Question

Let $g(x)$ and $h(x)$ be two polynomials of degree $d$ and $l$, respectively over a finite field $\mathbb{F}$. Then $\mathrm{Res}(g(x),h(x)) = 0$ iff $\deg(\gcd(g(x),h(x)) )> 0$.

Here is what I am trying.

If $\deg(\gcd(g,h))>0$ then Resultant$(g,h) = 0$.

If $\deg(\gcd(g,h))>0$, then there exist $a$ and $b$ such that $ag + bh = 0$ then I am going to get $d+l$ many homogenous equations with $d + l$ many unknowns so solution is unique if determinant of matrix that corresponds to homogenous equation is $0$.

I don't know how to prove the other direction.

Answer 1

You can follow along the euclidean algorithm to show the equivalence :

If $g = qh+r$ with $\deg(r) \le \deg(h)$, and $lc(h)$ denotes the leading coefficient of $h$,
then you can do row manipulations on the determinant to show that $Res(g,h) = (-1)^{\deg(g)\deg(h)}. lc(h)^{\deg(g)-\deg(r)}. Res(h,r)$,
so $Res(g,h) = 0 \iff Res(h,r) = 0$.

Repeat this step until you reach
$Res(\gcd(g,h),0)$, which is just $0^{\deg(\gcd(g,h))}$.
This is $1$ if the gcd has degree $0$ and $0$ otherwise.

In fact, you can be a bit more precise and show that the dimension of the kernel of the corresponding matrices are equal to the degree of the gcd.