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Here we consider the image of the principal minor map of symmetric matricesover an arbitrary unique factorization domain $R$. By exploiting a connectionwith symmetric determinantal representations, we characterize the image of theprincipal minor map through the condition that certain polynomials coming fromso-called Rayleigh differences are squares in the polynomial ring over $R$. Inalmost all cases, one can characterize the image of the principal minor mapusing the orbit of Cayley's hyperdeterminant under the action of $(SL_2(R))^{n}\rtimes S_{n}$. Over the complex numbers, this recovers a characterization ofOeding from 2011, and over the reals, the orbit of a single additionalquadratic inequality suffices to cut out the image. Applications to othersymmetric determinantal representations are also discussed.

Characterizing principal minors of symmetric matrices via determinantal multiaffine polynomials