The Geometry of Polynomials

Matt Boelkins Professor of Mathematics Grand Valley State University

In the geometry of polynomials, we seek to understand relationships among certain sets connected to polynomial functions. For instance, given a member of a family of polynomials, we may be interested in how the set of critical numbers is related to the corresponding set of the polynomial's zeros, with the goal of making general observations that apply to the entire family regarding the relationship between these sets. This subject area goes back at least to C. F. Gauss (1777-1855); Morris Marden (1905-1991) popularized its study through his classic text (1966).

In this talk, we'll discuss several fundamental historical results from the geometry of polynomials, including the Gauss-Lucas Theorem and Marden's Theorem. We will also survey some recent developments centered on the idea of polynomial root-dragging, the study of how continuously changing one or more roots of a polynomial function affects various properties of the function. Along the way, we'll consider a few results proved by undergraduate students and witness beautiful interplay between Euclidean geometry and calculus in the context of cubic polynomials.

This colloquium is accessible to students who've taken one semester of calculus.

Refreshments precede the talk at 3:30pm in NH-282.

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