Thesis Format

Integrated Article

Degree

Doctor of Philosophy

Program

Statistics and Actuarial Sciences

Sendov, Hristo

Abstract

This work investigates several geometric properties of the solutions of the multi-affine polynomials. Chapters 1, 2 discuss two different notions of invariant circles. Chapter 3 gives several loci of polynomials of degree three. A locus of a complex polynomial p(z) is a minimal, with respect to inclusion, set that contains at least one point of every solution of the polarization of the polynomial. The study of such objects allows one to improve upon know results about the location of zeros and critical points of complex polynomials, see for example [22] and [24]. A complex polynomial has many loci. It is natural to look for loci with special properties. For example finding a locus inside the smallest closed disk containing the zeros of p, see [21, Theorem 5.1] or a locus with the smallest area.

In Chapter 1, we investigate a connection between certain (skew-)coninvolutory matrices, multi-affine polynomials and n-tuples of circles in the complex plane, that are invariant under such polynomials. It is well-known that any non-degenerate Mobius transformation T(z) = (az+b)/(cz+d), sends circles into circles. If for any z_1 in the extended complex plane, one lets z_2:=T(z_1), then trivially the pair (z_1,z_2) is a solution of the bi-affine polynomial P(z_1,z_2):=cz_1z_2-az_1+dz_2-b. We investigate a natural generalization of these observations and consider a multi-affine polynomial P(z_1,..., z_n) of degree n. We say that the n-tuple of circles (C_1,..., C_n) in the extended complex plane are invariant with respect to P, if for any k = 1,...,n, and any z_i on C_i, i is not k, there exists a z_k on C_k, such that (z_1,...0,z_n) is a solution of P. Given an n-tuple of circles (C_1,..., C_n), we give two characterizations of all multi-affine polynomials that preserve them. The opposite problem: given a multi-affine polynomial, find all n-tuples of circles that are invariant with respect to that polynomial, turns out to be much harder. We answer the opposite question only for multi-affine symmetric polynomials. It turns out that very few symmetric multi-affine polynomials (only the polarization of polynomials whose zeros are symmetric with respect to a circle) have invariant circles. So, the notion of invariant circles is a relatively strong property.

In Chapter 2, we introduce a relaxation of the notion of invariant circles in Chapter 1, called weakly invariant circles with respect to the polynomial given a solution. The main results, give necessary and sufficient conditions for weakly invariant circles given a solution to exist and describes how to find such circles. In addition, we show that if a non-trivial, symmetric, multi-affine polynomial has weakly invariant circles given a solution (u_1,...,u_n), with distinct points, then the points u_1,...,u_n are necessarily on a circle.

In Chapter 3, we give a simpler and more transparent proof of [21, Theorem~5.1], which constructs a locus of a polynomial of degree three inside the smallest disk containing its zeros. The original proof relies on intricate geometric constructions and relies on a specific positioning of the zeros of the polynomial. Then, we extend this result by constructing a locus holder of the polynomial, when its zeros are in arbitrary position. Finally, we answer an open question formulated at the end of Section 7 in [21] and by doing so discover a new locus of the polynomial z^3+1.

Summary for Lay Audience

The location of the zeros of a univariate polynomial has always been a fundamental and important problem in mathematics. Many problems are related to polynomials with special location of the zeros. The classical Grace-Walsh-Szego Coincidence Theorem describes a connection between the location of the zeros of a polynomial and the solutions of a unique symmetric multi-affine polynomial corresponding to the polynomial. In other words, the geometric properties of the solutions of that symmetric multi-affine polynomial help us to localize the zeros of the polynomial.

This work investigates several geometric properties of the solutions of the multi-affine polynomials, not necessarily symmetric. We introduce the notion of invariant n-tuple of circles: if the first n-1 points from a solution are on the first n-1 circles, then the n-th point from that solution is always on the n-th circle. Given any n circles, we find all multi-affine polynomials that preserve them. However, not all multi-affine polynomials have invariant circles. In fact, very few symmetric multi-affine polynomials have invariant circles. So, the notion of invariant circles is a relatively strong property. We then introduce a relaxation of the notion of invariant circles, called weakly invariant circles given a solution. We give necessary and sufficient conditions for weakly invariant circles to exist. It turns out that for most polynomials, if they have weakly invariant circles given a solution with distinct points, then the points from the solution are necessarily on a circle. The last part of this thesis gives a simpler proof of [21, Theorem~5.1], which constructs a locus of a polynomial of degree three inside the smallest disk containing its zeros, and answers an open question formulated at the end of Section~7 in [21].