Characterizing principal minors of symmetric matrices via determinantal multiaffine polynomials (Q6066482)

Given an \(n \times n\) matrix \(A\) with entries in a commutative ring \(R\), let \(A_S\) denote the principal minor obtained by taking the determinant of the principal submatrix of \(A\) with rows and columns indexed by \(S\). Let \(\operatorname{Sym}_2(R^n)\) denote the space of symmetric \(n \times n\) matrices with entries in \(R\). The principal minor map is the map \[ \pi : \operatorname{Sym}_2(R^n) \to R^{2^n} \text{ given by } \pi(A) = (A_S)_{S \subseteq [n]}, \] where we take \(A_\emptyset = 1\). \textit{O. Holtz} and \textit{B. Sturmfels} [J. Algebra 316, No. 2, 634--648 (2007; Zbl 1130.15005)] showed that the image of the principal minor map is invariant under an action of \(\operatorname{SL}_2(\mathbb{R})^n \rtimes S_n\) and conjectured that the vanishing of polynomials in the orbit of the hyperdeterminant under this group cuts out the image of the principal minor map over \(\mathbb{C}\). This was resolved by \textit{L. Oeding} [Algebra Number Theory 5, No. 1, 75--109 (2011; Zbl 1238.14035)], and the authors of this paper extend this result to hold over arbitrary unique factorization domains, except those with exactly three elements. To study the principal minor map, they associate the image \((A_S)_{S\subseteq [n]}\) with the coefficients of \(\operatorname{det}(\operatorname{diag}(x_1, \dots, x_n) + A) = \sum_{S \subseteq [n]} A_S \prod_{i \notin S} x_i.\) This translates the problem of characterizing the image of the principal minor map to the problem of characterizing multiaffine polynomials in \(R[x_1, \dots, x_n]\) with symmetric determinantal representations. Through the Rayleigh difference of polynomials, one can characterize multiaffine polynomials in \(R[x_1, \dots, x_n]\) with symmetric determinantal representations. Then, the authors characterize the image of the principal minor map by checking the Rayleigh difference of the associated multiaffine polynomial is a square: Theorem 3.5. Let \(R\) be a unique factorization domain. An element \(\mathbf{a} = (a_S)_{S \subseteq [n]}\) in \(R^{2^n}\) is in the image of \(\operatorname{Sym}_2(R^n)\) under the principal minor map if and only if \(a_{\emptyset} = 1\) and for every \(i, j \in [n]\), \(\Delta_{ij}(f_{\mathbf{a}})\) is a square in \(R[x_1, \dots, x_n]\). A necessary condition for being a quadratic polynomial a square is that the discriminant is zero. However, they prove that the condition is sufficient for characterizing the image of the principal minor map. Theorem 5.1. Let \(R\) be a unique factorization domain with \(|R| = 3\), and \(\mathbf{a} = (a_S)_{S \subseteq [n]} \in R^{2^n}\) with \(a_\emptyset = 1\). There exists a symmetric matrix over \(R\) with principal minors if and only if \begin{itemize} \item[1.] for every \(i, j \in [n]\), \(a_ia_j - a_{ij}\) is a square in \(R\), and \item[2.] for every \(\gamma \in \operatorname{SL}_2(R)^n \rtimes S_n\), \((\gamma \cdot \operatorname{HypDet})(a) = 0\), where \(\operatorname{HypDet}(a)\) is the Cayley \(2 \times 2 \times 2\) hyperdeterminant. \end{itemize} As additional outcomes, they recover Oeding's result over \(\mathbb{C}\) and provide a semialgebraic description of the image of the principal minor map over \(\mathbb{R}\).