Determinantal Representations of Polynomials

I shall discuss the problem of representing a given polynomial $p \in {\mathbb C}[x_1,\ldots,x_d]$ as the determinant of a matrix of affine linear forms, i.e., <br/>$$p(x_1,\ldots,x_d) = \det(A_0 + x_1 A_1 + \cdots + x_d A_d),$$ <br/>where $A_0,A_1,\ldots,A_d$ are $N \times N$ complex matrices for some $N \geq \deg p$. This problem, especially its real symmetric and positive versions, are both interesting by themselves and appear in a variety of questions in operator theory, function theory, hyperbolic PDEs, etc. <br/> <br/>In case $N = \deg p$ the problem converts naturally to a similar representation problem for homogeneous polynomials, which can then be treated using the tools of classical algebraic geometry. This leads to a very complete and explicit picture in case $d=2$. <br/> <br/>For $d&gt;2$ the homogeneous problem has in general no solutions, i.e., necessarily $N &gt; \deg p$. I shall describe an approach based on noncommutative techniques where we first lift the given polynomial $p$ to a polynomial in noncommuting variables, and then tackle the corresponding representation problem for noncommutative polynomials using some recent ideas coming from realization theory for noncommutative rational functions. <br/> <br/>I shall also discuss the representation problem for real polynomials with real symmetric matrices, especially with positive definite ones. This leads to the so called hyperbolic polynomials, to a 1958 conjecture of Peter Lax which has been recently established, and to its possible generalizations. <br/> <br/>Most of this talk is based on joint work with Joe Ball, Bill Helton, and Scott McCullough.