Grant: Discovery Grants Program - Individual

The classical theorem of Grace states that every circular domain in the complex plane containing the zeros of a polynomial p(z), contains a zero of any of its apolar polynomials. Recently, we introduced the notion of a locus of a complex polynomial p(z). It is a smallest (with respect to inclusion) closed set in the complex plane that contains a zero of any of its apolar polynomials. We established several general properties of the loci and showed, in particular, that the property of a set being a locus of a polynomial is preserved under a Mobius transformation, that every locus is the closure of its interior, and that every locus is the closure of the zeros of all polar derivatives of p(z) having poles outside of the locus. We also showed the connection between the notion of the locus and several other classical theorems on Geometry of Polynomials, such as Laguerre's theorem, Rolle's theorem, Grace-Szego-Walsh Coincidence theorem. In every instance the notion of a locus provides a minimal set for which each one of these theorems holds. The class of all loci of a polynomial is a very rich having intriguing properties. Our first goal is to shed light on these properties, isolate and investigate several subclasses of loci, such as the locus with the smallest area, loci with smooth boundary, or loci with symmetries. Another goal is to develop computationally efficient algorithms for the (approximate) computation of a locus of a polynomial, also known as a locus holder. The connections between a locus and the zeros of the polar derivatives of a polynomial suggests an approach for attacking the Bl. Sendov's Conjecture dating back to 1962.*Another research direction that I am pursuing concerns the properties of spectral functions (SpF). A SpF is a function on a Hermitian matrix argument, invariant under unitary similarity transformation. Each SpF can be represented in a unique way as the composition of a symmetric function on R^n with the eigenvalues of a Hermitian matrices. These functions find numerous applications in diverse areas such as classical complex analysis, optimization, non-smooth and matrix analysis, elasticity, statistics, and quantum physics. If, in particular, the symmetric function on R^n is separable, then we obtain the so-called separable SpF. Through their derivative, the latter are connected with the primary matrix functions finding applications in the theory of operator monotone and operator convex functions. The known connection between the SpF's and the primary matrix functions is unsatisfactory. I am currently working on a family of matrix-valued maps that generalizes the SpF's and the primary matrix functions. This allows the entire theory of SpF's and the primary matrix functions to be extended, and in particular leads to a generalization of the classical notions of operator monotone and operator convex functions.*