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Consider the ring $\mathbb{Q}[x_1,\ldots,x_n]$ of multivariate polynomials with rational coefficients. Given a positive integer $d$, let $\mathcal{C}_d \subset \mathbb{Q}[x_1,\ldots,x_n]$ be the class of polynomials $f$ which can be written as

$$f = \operatorname{det}(A)$$

where $A$ is a $d$-by-$d$ matrix with entries which are affine in the $x_1, \ldots, x_n$ with rational coefficients. It is of course possible to consider entries which are simply in $\mathbb{Q}[x_1,\ldots,x_n]$, but in the problem I am interested in, I think it is enough to consider entries that are affine in the $x_1, \ldots, x_n$.

If $f \in \mathcal{C}_d$, let us say that $f$ has a degree $d$ determinantal representation (with affine entries).

Question 1: concretely, how to devise an algorithm to tell whether or not a polynomial $f \in \mathbb{Q}[x_1,\ldots,x_n]$ of degree $d$ is actually in $\mathcal{C}_d$ or not?

Question 2: assuming there exists at least one such determinantal representation, corresponding to some $d$-by-$d$ matrix $A$, then there are infinitely many other representations, since for instance we could replace $A$ with $PAQ$, where $P,Q \in \mathrm{SL}(d,\mathbb{Q})$. Is there a way to concretely devise an algorithm that would give the "nicest" such representation? For example, if there is a representation $A$ with entries that are affine with integer coefficients, then this is "nicer" than a representation with rational coefficients which are not integers. It is ambiguous, I agree, but if someone could make this more precise, either in the comments or as an answer, then I would appreciate it.

Malkoun
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  • for cubic $g(x,y,z)$ in three variables, we take the Hessian matrix, the elements are linear in the three variables. Then, take the determinant of that, call it $h(x,y,z).$ The original $g$ factors completely into linear factors if and only if $h$ is a constant multiple of $g.$ See example https://math.stackexchange.com/questions/2678486/how-to-show-that-if-x-y-z-are-rational-numbers-satisfying-x-y-z3-9/2678675#2678675 – Will Jagy Mar 13 '21 at 23:29
  • @WillJagy, thank you for your comment and answer. Well, I am interested in such determinantal representations (when they exist) to understand better the vanishing locus of the given polynomial. The result you have quoted is interesting though. I did not know it. But my interest in such representations is a bit from a different perspective. – Malkoun Mar 13 '21 at 23:49
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    At least in the real case, there are interesting results published in the last 10 years or so: you might find it helpful to start with Polynomials with and without determinantal representations. – KReiser Mar 14 '21 at 01:02
  • @KReiser, this does look interesting! Thank you. I thought the problem was natural, and I was sure it was considered before. – Malkoun Mar 14 '21 at 01:28

2 Answers2

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In https://arxiv.org/abs/1804.00145, there is an algorithm which explains how to write any polynomial $f \in \mathbb{Z}[x_1,\ldots,x_n]$ as

$$ f = \operatorname{det}(A),$$

where $A$ is matrix whose entries are affine in the variables $x_i$. Actually, it is even possible to choose $A$ so that in any column of $A$, at most one of the variables $x_i$ "appears".

I also refer to the related post https://mathoverflow.net/questions/385566/expressing-a-polynomial-as-the-determinant-of-a-matrix-of-linear-forms#comment985383_385566 and the discussion and comments there.

I wonder who first proved the result above, that any $f \in \mathbb{Z}[x_1,\ldots,x_n]$ has a determinantal representation.

Malkoun
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One result from an encyclopedic book by Schinzel. He has his own Hypothesis. You seem to be talking about complete factorization, which is more restrictive, while the tests get simpler to describe.

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Will Jagy
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