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This is an irresponsible question: I do not have done any thinking on it, or even literature search.

I just became curious whether there is some modification of the notion of a common root of two polynomials which would be detected by a Pfaffian of some alternate matrix, rather than a determinant of some general matrix, like it is the case with the resultant.

(PS Seems like there is no tag "algebra", so I chose commutative algebra instead)

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    $\begingroup$ Another possible question: the Determinant is an irreducible polynomial. But when restricted to alternate matrices, it becomes the square of something (which is the Pfaffian). Likewise, the Resultant is an irreducible polynomial. Does is factorize when one restrict to an interesting subspace of $k[X]$ ? $\endgroup$
    – Denis Serre
    Sep 26, 2017 at 7:37
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    $\begingroup$ @DenisSerre That seems worthy of asking as a separate question. $\endgroup$
    – j.c.
    Sep 26, 2017 at 18:30
  • $\begingroup$ Denis Serre asked his question here: mathoverflow.net/questions/282069/… $\endgroup$
    – j.c.
    Sep 26, 2017 at 23:00

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Pfaffian resultant formulas are obtained in Resultants and Chow forms via Exterior Syzygies (2001), where the polynomials are represented by coordinates on a Grassmanian manifold.

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    $\begingroup$ This is a fascinating paper indeed, although so far I could not extract any concise answer from it. I admit there are lots of very interesting examples. Seems like Chow forms of rank 2 vector bundles are Pfaffians, but I could not reduce this to anything simpler. Still, I think I will accept this answer... $\endgroup$
    – მამუკა ჯიბლაძე
    Sep 26, 2017 at 11:08

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