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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 $$n=2,$$ $$n=3 \quad \textrm{and} \quad d \leq 15 \quad \textrm{or}$$ $$n=4 \quad \textrm{and} \quad d \leq 5.$$

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 $$n=2,$$ $$n=3 \quad \textrm{and} \quad d \leq 15 \quad \textrm{or}$$ $$n=4 \quad \textrm{and} \quad d \leq 5.$$

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 either $$n=2 \quad \textrm{or}$$ $$n=d=3.$$

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

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 $$n=2,$$ $$n=3 \quad \textrm{and} \quad d \leq 15 \quad \textrm{or}$$ $$n=4 \quad \textrm{and} \quad d \leq 5.$$

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:

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

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

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 either $$n=2 \quad \textrm{or}$$ $$n=d=3.$$

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