On the Geometry of Multi-Affine Polynomials
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.