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Books > Science & Mathematics > Mathematics > Algebra
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Polynomials, Dynamics, and Choice - The Price We Pay for Symmetry (Paperback)
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Polynomials, Dynamics, and Choice - The Price We Pay for Symmetry (Paperback)
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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.
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