## Graduate Student Theses, Dissertations, & Professional Papers

2020

Dissertation

#### Degree Type

Doctor of Philosophy (PhD)

Mathematics

#### Department or School/College

Department of Mathematical Sciences

Jennifer Brooks

#### Commitee Members

Karel Stroethoff, Elizabeth Gillaspy, Kelly McKinnie, Eijiro Uchimoto

mathematics

#### Publisher

University of Montana

#### Abstract

We study a natural extremal problem about the vector space consisting of all homogeneous polynomials of degree $d$ in $n+1$ variables with real coefficients, together with the zero polynomial. We define the {\it rank} of a polynomial to be the number of distinct monomials appearing in the polynomial with non-zero coefficient. We are particularly interested in those homogeneous polynomials whose quotient with the homogeneous polynomial $x_0+x_1+\cdots+x_n$ is a polynomial of degree $d-1$ with maximal rank. For each degree $d$, we seek the minimum rank for an element of this subfamily and we seek to describe those polynomials with minimum rank. We call such polynomials {\it sharp} polynomials.

These problems have a simple solution for polynomials in one and two variables. The three-variable case is interesting and non-trivial, but well-understood. This research question has its roots in the study of proper polynomial mappings between balls in complex Euclidean spaces of different dimensions and the degree estimates problem. D'Angelo, Kos and Riehl \cite{degree} and Lebl and Peters \cite{hyperplaneq} used a graph-theoretic approach to solve this problem in the case of proper monomial mappings. Lebl and Peters give a minimum rank estimate that answers our question in the three variable case. A family of sharp polynomials was described by D'Angelo and has been extensively studied. Brooks and Grundmeier \cite{notes} provided a new proof of the minimum rank theorem in the three-variable case using a commutative algebra approach. They reformulate the problem as a question about homogeneous ideals and address it by studying the Hilbert function and the graded Betti number of certain ideals.

Using the same method as Brooks and Grundmeier, we give a sharp estimate for the minimum rank of homogeneous polynomials of our subfamily in four variables as well as a family of sharp polynomials. Moreover, we state a general result on the minimum rank for polynomials in $n+1$ variables. Although this estimate is sharp in the three- and four- variable cases, the estimate is not sharp when the number of variables is greater than four.

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