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


The pfaffian can be defined as $\sqrt{{\rm det}(A) } $ when $A$ is skew symmetric, or explicitly $${\rm pf}(A) = \frac{1}{2^n n!}\sum\limits\_{\sigma \in S\_{2n}}{\rm sgn} (\sigma)\prod\limits\_{j=1}^n a\_{\sigma (2j -1 ),\sigma(2j)}.$$ 



Detecting if a polynomial is a Pfaffian
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](http://www-math.unice.fr/publis/beauvill_det.ps), 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 $$n=2,$$ $$n=3 \quad \textrm{and} \quad d \leq 15 \quad \textrm{or}$$ $$n=4 \quad \textrm{and} \quad d \leq 5.$$


