Polynomials, Dynamics, and Choice: The Price We Pay for Symmetry

Working  out solutions to polynomial equations is a mathematical problem that  dates from antiquity. Galois developed a theory in which the obstacle to  solving a polynomial equation is an associated collection of  symmetries. Obtaining a root requires "breaking" that symmetry. When the  degree of an equation is at least five, Galois Theory established that  there is no formula for the solutions like those found in lower degree  cases. However, this negative result doesn't mean that the practice of  equation-solving ends. In a recent breakthrough, Doyle and McMullen  devised a solution to the fifth-degree equation that uses geometry,  algebra, and dynamics to exploit icosahedral symmetry.

Polynomials,  Dynamics, and Choice: The Price We Pay for Symmetry is organized in two  parts, the first of which develops an account of polynomial symmetry  that relies on considerations of algebra and geometry. The second  explores beyond polynomials to spaces consisting of choices ranging from  mundane decisions to evolutionary algorithms that search for optimal  outcomes. The two algorithms in Part I provide frameworks that capture  structural issues that can arise in deliberative settings. While  decision-making has been approached in mathematical terms, the novelty  here is in the use of equation-solving algorithms to illuminate such  problems.

Features

• Treats  the topic―familiar to many―of solving polynomial equations in a way  that’s dramatically different from what they saw in school
• Accessible to a general audience with limited mathematical background
• Abundant diagrams and graphics.