# Polynomial with two repeated roots

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.

-
The GCD of the derivative $f'$ with $f$ has roots precisely where $f$ has repeated roots. – Douglas Zare Feb 1 '10 at 13:54
Indeed it easy to find one repeated root. My problem was to detect polynomials with two repeated roots. But that turned out to be easy nevertheless; see David's answer. – Andrea Ferretti Feb 1 '10 at 14:15
What do you mean only one root? When there are two repeated roots, the GCD is a quadratic with roots at the repeated roots. For example, $GCD(x^4 - 2x^2 + 1, 4x^3 - 4x) = x^2-1 = (x-1)(x+1)$. – Douglas Zare Feb 1 '10 at 14:28
Yes, but this condition is complicated to write down explicitly in terms of the coefficients of $f$, because you have to track down what happens when you perform the Euclidean algorithm. It may be feasible to find equations for the locus in this way, but I don't think that computations are significantly simpler than equating f to a generic square (but I may be wrong). – Andrea Ferretti Feb 1 '10 at 14:53

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 $4e=c^2$.