In this talk, I shall give a panoramic view of my research work. I shall introduce the notion of hyperbolic polynomials and discuss an algebraic method to test hyperbolicity of a multivariate polynomial w.r.t. some fixed point via sum-of-squares relaxation, proposed in my research work. An important class of hyperbolic polynomials are definite determinantal polynomials. Heltonâ€“Vinnikov curves in the projective plane are cut-out by hyperbolic polynomials in three variables. This leads to the computational problem of explicitly producing a symmetric positive definite linear determinantal representation for a given curve. I shall focus on two approaches to this problem proposed in my research work: an algebraic approach via solving polynomial equations, and a geometric-combinatorial approach via scalar product representations of the coefficients and its connection with polytope membership problem. The algorithms to solve determinantal representation problems are implemented in Macaulay2 as a software package DeterminantalRepresentations.m2. Then I shall briefly address the methodologies to find the degree and the defining equations of certain varieties which are obtained as the image of some given varieties of $\mathbb{P}_n$ under coordinate-wise power map, for example the $4 \times 4$ orthostochastic variety. Finally, I shall demonstrate a connection of symmetroids with the real degeneracy loci of matrices.

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- Seminars for 2021

Last updated: 18 Sep 2024