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

Galois Theory states that there is no formula for the solutions to polynomial equations of degree five or higher, but Doyle and McMullen have devised a solution to the fifth-degree equation that uses geometry, algebra, and dynamics to exploit icosahedral symmetry.

Format: Hardback
Length: 169 pages
Publication date: 23 August 2022
Publisher: Taylor & Francis Ltd

Solving polynomial equations is a mathematical problem that has existed since 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. Galois Theory established that there is no formula for the solutions like those found in lower degree cases when the degree of an equation is at least five. However, this negative result does not 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 is dramatically different from what they saw in school.
• Accessible to a general audience with limited mathematical background.
• Abundant diagrams and graphics.

Weight: 510g
Dimension: 234 x 156 (mm)
ISBN-13: 9780367565206