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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.

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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
up vote 7 down vote accepted

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 $4e=c^2$.

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uhm... sorry! I tried some computations without making the easy reduction and soon began thinking there should be a more conceptual way, so I asked here. I should have tried harder before posting. anyway thank you for your quick reply! – Andrea Ferretti Feb 1 '10 at 14:03
What about for a more general problem: Given a degree n polynomial, can we decide if it has r roots appearing with given multiplicities? – mdeland Feb 3 '10 at 18:10
One can ask a similar question for any configuration of multiple roots encoded by an integer partition. Set theoretic equations for these general coincident root loci can be found in this paper by Chipalkatti As for the set theoretical equations, this very much open. – Abdelmalek Abdesselam Jul 12 '10 at 0:57
I meant to say scheme theoretic equations in the last sentence. – Abdelmalek Abdesselam Jul 12 '10 at 20:29

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