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When is a degree-$n$ homogeneous polynomial in $\mathbb{C}[x_1, x_2, \ldots , x_m ]$ the product of $n$ one-forms?

Is there any simple algorithm or criterion to check it?

I have chosen the complex field to avoid any field insufficiency.

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    $\begingroup$ Does this answer your question? which homogeneous polynomials split into linear factors? $\endgroup$ Commented May 9 at 6:39
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    $\begingroup$ @GjergjiZaimi Not obvious to me that it does. The current question is specifically about sub-question 4. of that question, and it looks like that sub-question was left unanswered. $\endgroup$
    – R.P.
    Commented May 9 at 7:34
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    $\begingroup$ This is set theoretically defined by Brill's equations for the Chow variety, although the ideal they generate is not in general radical. $\endgroup$ Commented May 9 at 12:21
  • $\begingroup$ I think i did answer that question. $\endgroup$ Commented May 9 at 17:33
  • $\begingroup$ @R.P.: Q4 is rather moot. As far as I know, the simplest criterion is the answer to Q3, i.e., to just plug the coefficients of the polynomial of interest into the Brill-Gordan equations and see if one gets zero or not. $\endgroup$ Commented May 9 at 19:37

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