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Given cubic polynomial $a x^3 + b x^2 + c x + d = 0$, with potentially complex coefficients, is there a sufficient condition for two of the roots having equal imaginary part? That is, is there a condition on the coefficients that allows me to guarantee that the roots take the form $s + i t$, $u + i t$, and $v + i w$?

Specifically, I am looking for a condition analogous to the cubic discriminant (i.e. condition for a cubic to have a repeated root).

Paul Ullrich
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  • https://en.wikipedia.org/wiki/Cubic_function – Gregory May 22 '17 at 14:51
  • As you said use the discriminant to determine what conditions are needed. What do you mean equal imaginary part? are you saying that they will be of the form $a \pm bi$ or you want $a + bi$ and $c + bi$? – Gregory May 22 '17 at 14:52
  • $a+bi$ and $c+bi$, not to be confused with the coefficients in the cubic. For instance, if the roots are 6-7i, -17+5i and 3+5i the cubic would qualify. – Paul Ullrich May 22 '17 at 15:07
  • For starters, you would have to have at least one coefficient be complex. Aside from that obvious fact, you probably just need to go through the motions of manipulating the discriminant and it will probably be messy. – Gregory May 22 '17 at 19:37

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