I have a polynomial of degree 4 $f(t) \in \mathbb{C}[t]$, and I'd like to know when it has two repeated roots, in terms of its coefficients.
Phrased otherwise I'd like to find the equations of the image of the squaring map
$sq \colon \mathbb{P}(\mathbb{C}[t]^{\leq 2}) \rightarrow \mathbb{P}(\mathbb{C}[t]_{\leq 4})$.
(for some reason the first lower index wouldn't parse, so I put it on top).
Of course I can write the map explicitly and then find enough equations by hand, but this looks cumbersome. I'm not an expert in elimination theory, so I wondered if there is some simple device to find explicit equations for this image. For instance one can detect polynomials with one repeated root using the discriminant, but I don't know how to proceed from this.
Best Answer
It seems to me that this example is easy to do by hand. By the standard tricks, we can assume your polynomial is of the form $$x^4+ c x^2 + dx +e.$$ A polynomial of this form is a square if and only if $d=0$ and $4e=c^2$.