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UCCS MATHEMATICS COLLOQUIUM

Heights of Polynomials and Random Matrix Theory

Abstract: A height of a polynomial is a measure of its complexity. One example of a height is Mahler's measure, defined to be the absolute value of the leading coefficient of a polynomial times the product of the moduli of its roots outside the unit circle. As a function on coefficient vectors of degree N polynomials, Mahler's measure satisfies all of the axioms of a vector norm except the triangle inequality. In this setting, the unit ball of Mahler's measure can be used to discover information about the range of Mahler's measure on integer polynomials. This volume can also be interpreted in the setting of random matrix theory. We will discuss this surprising connection and how it leads to new results in random matrix theory.