Determinantal representations and the Hermite matrix (Q357531)

A polynomial \(p\in \mathbb{R}[x_1,\dots,x_n]\) with \(p(0)=1\) is called a real-zero polynomial if \(p\) has only real zeros along every line through the origin. For example, real-zero polynomials are polynomials with determinantal representation NEWLINE\[NEWLINEp=\det(I+A_1x_1+\dots+A_nx_n),NEWLINE\]NEWLINE where \(A_i\) are symmetric \(k\times k\) matrices.NEWLINENEWLINEThe naturals question is whether every real-zero polynomial has a determinantal representation. The answer is positive in the case of two variables and negative in higher dimensions. Nevertheless, such a representation exists for some power of the polynomial \(p\).NEWLINENEWLINEIt is natural to ask also in the case when the determinantal representation exists what is the size \(k\) of the matrices \(A_i\). And it is desirable to obtain the determinantal representation explicitly.NEWLINENEWLINEThe main results of the paper are the following. With each polynomial \(p\), an Hermite matrix \(\mathcal{H}(p)\) is associated. It is proved that a determinantal representation of some power of \(p\) of the correct size gives a sum-of-squares representation of \(\mathcal{H}(p)\). Then it is proved that if \(\mathcal{H}(p)=Q^tQ\) then a definite determinantal representation of a multiple of \(p\) can be found if a certain extension problem for linear maps on free graded modules derived from \(Q\) has a solution. Finally, it is proved that in the case when \(\mathcal{H}(p)\) is positive semidefinite (and thus has a sum-of-squares representation with denominators), there exists a determinantal representation with denominators.