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In my previous question asking about the co-intersection of three circles, a degree six polynomial in twelve variables was found for a special case. This polynomial has precisely 720 terms, of which exactly half are positive and half are negative.

This makes me strongly suspect there is a 6x6 matrix lurking behind the scenes whose determinant is that polynomial. This leads me to the question: How can you tell if an expression is the determinant of a matrix, and if so, how can you find that matrix?

It would be particularly helpful if it could be done for the special case of the polynomial from my previous problem.

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    $\begingroup$ Well, every polynomial is a determinant in a very trivial way. Presumably you want a matrix whose entries are "relatively simple" compared to the polynomial. $\endgroup$
    – Qiaochu Yuan
    Commented Jun 22, 2022 at 2:08
  • $\begingroup$ Yeah, I would like the entries to be simple. Ideally, given that the polynomial from the previous question is of degree 6, and the number of terms is 6 factorial (720), then the matrix would be 6x6 with all terms corresponding to one of the coordinates (where there will be some repetition) $\endgroup$
    – Thomas Blok
    Commented Jun 22, 2022 at 5:26

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