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Given an explicit polynomial, is there any kind of trick/algorithm to check whether it is a pfaffian of a matrix with linear entries?

edited Jun 26 2011 at 17:42

asked Jun 25 2011 at 16:41

IMeasy
1,355●4●12

	What is a Pfaffian polynomial? – Qiaochu Yuan Jun 25 2011 at 17:26
	Please try to improve the question! As it stands, it might (and probably should) get closed for lack of clarity, motivation etc. – Gjergji Zaimi Jun 26 2011 at 3:48
	As motivation for related questions, Mumford gives, in his first paper on Prym varieties, an explicit Pfaffian of linear forms, which defines the tangent cone to the theta divisor of a Prym variety precisely when that Pfaffian is not identically zero. When this occurs is more complicated however and was only settled some 25 years later. – roy smith Jun 26 2011 at 15:51
	I am sorry for the sloppyness of the question. I assumed, without stating, that the matrix had linear entries. Now I've edited it. – IMeasy Jun 26 2011 at 17:43

Francesco Polizzi · Answer 1 · 2011-06-25 21:39:16Z UTC

As Bruce Westubury noticed, the answer to this question is trivial as it is stated.

Surprisingly enough, however, the situation becomes very interesting when one considers representations of homogeneous polynomials as pfaffians of matrices with linear entries.

More precisely, let us consider the following version of the question:

QUESTION. Let $F \in k[x_0, \ldots, x_n ]$ be a homogeneous polynomial of degree $d$. Does there exist a symmetric [resp. antisymmetric] matrix $M$, whose entries are linear forms, such that $$\det(M)=F \quad (\textrm{resp}. \ \textrm{Pf}(M)=F)?$$

This problem is studied in detail in Beauville's paper [Symmetric determinantal hypersurfaces, Michigan Mathematical Journal 48 (2000)], where the existence of a symmetric or pfaffian representation is related to the existence of certain vector bundles on the projective variety $X \subset \mathbb{P}^n_k$ defined by $F=0$.

Among other things, Beauville proves the following results (when $k= \mathbb{C}$):

$\bullet$ A general polynomial of degree $d$ admits a determinantal representation if and only if $$n=2 \quad \textrm{or}$$ $$n=3 \quad \textrm{and} \quad d\leq 3.$$

$\bullet$ A general polynomial of degree $d$ admits a pfaffian representation if and only if $$n=2,$$ $$n=3 \quad \textrm{and} \quad d \leq 15 \quad \textrm{or}$$ $$n=4 \quad \textrm{and} \quad d \leq 5.$$

Bruce Westbury · Answer 2 · 2011-06-25 17:06:12Z UTC

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The Pfaffian of $\begin{pmatrix} 0 & A \\ -A & 0\end{pmatrix}$ is $\det A$. Since any polynomial is a determinant this means any polynomial is a Pfaffian.

answered Jun 25 2011 at 17:06

Bruce Westbury
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Hailong Dao · Answer 3 · 2011-06-26 20:10:13Z UTC

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Now that the question makes sense, let me mention the following result (you can find it in Eisenbud, Exercise 20.17) which hinted at what Francesco wrote.

Let $R = k[[x,y,z]]$ and $f\in m=(x,y,z)$. Then $f$ is a determinant of a matrix (of size at least $2$) with entries in $m$ iff $R/(f)$ is not a UFD!

edited Jun 26 2011 at 20:10

answered Jun 26 2011 at 19:14

Hailong Dao
15.4k●29●70

	Something is wrong here. Take $f=z$. Then $R/(f) = k[x,y]$ is a UFD, and $f = \det \ (z)$ . – David Speyer Jun 26 2011 at 19:52
	David, thanks. Of course the size of the matrix has to be bigger than one, I fixed it now. – Hailong Dao Jun 26 2011 at 20:05

Detecting if a polynomial is a Pfaffian

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