Determinantal Representation and the image of the principal minor map

Characterizing principal minors of symmetric matrices via determinantal multiaffine polynomials

Abeer Al Ahmadieh, Cynthia Vinzant

Here we consider the image of the principal minor map of symmetric matrices over an arbitrary unique factorization domain R. By exploiting a connection with symmetric determinantal representations, we characterize the image of the principal minor map through the condition that certain polynomials coming from so-called Rayleigh differences are squares in the polynomial ring over R. In almost all cases, one can characterize the image of the principal minor map using the orbit of Cayley's hyperdeterminant under the action of (SL2(R))^nxSn. Over the complex numbers, this recovers a characterization of Oeding from 2011, and over the reals, the orbit of a single additional quadratic inequality suffices to cut out the image. Applications to other symmetric determinantal representations are also discussed.

A generalization of the space of complete quadrics

Abeer Al Ahmadieh, Mario Kummer, Miruna-Stefana Sorea

To any homogeneous polynomial h we naturally associate a variety Ωh which maps birationally onto the graph Γh of the gradient map ∇h and which agrees with the space of complete quadrics when h is the determinant of the generic symmetric matrix. We give a sufficient criterion for Ωh being smooth which applies for example when h is an elementary symmetric polynomial. In this case Ωh is a smooth toric variety associated to a certain generalized permutohedron. We also give examples when Ωh is not smooth.