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Abstract

In this paper we consider the set of positive points at which a polynomial with positive coefficients, arbitrary dimension n, and real powers is equal to a fixed positive constant c. We find that when this set is non-empty and is bestowed with the relative Euclidean topology coming from R^n, it is homeomorphic to a codimension one piecewise linear set that depends only on the polynomial's powers. This piecewise linear set can, in a certain sense, be interpreted as a bijectively mapped version of the original set as the constant c approaches infinity. In addition to this result, we provide a condition on the polynomial powers for testing if the solution space is homeomorphic to the n-1 dimensional sphere S^(n-1), and derive piecewise linear inner and outer bounds for our solution set. These provide bounds in the sense that each point in our solution set that lies on any particular ray originating at the origin is trapped between a unique inner and outer bound point also on that ray. While this paper provides insight into only a specific type of polynomial slice, an appropriate generalization of these observations might one day lead to improved techniques for analyzing slices of high dimension polynomials in general, objects which appear frequently throughout mathematics.

Author Bio

I am a recent graduate of Columbia University's School of Engineering and Applied Science. I began the research that would lead me to the idea for this paper during the summer of 2004 as part of the VIGRE undergraduate research program at Columbia. During the following year I developed this idea further and it turned into my senior thesis. My current interests are applying mathematics to search engine design, text analysis, and finance.

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