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Polynomials are perhaps the most important family of functions in
mathematics. They feature in celebrated results from both antiquity
and modern times, like the insolvability by radicals of polynomials
of degree 5 of Abel and Galois, and Wiles' proof of Fermat's "last
theorem." In computer science they feature in, e.g.,
error-correcting codes and probabilistic proofs, among many
applications. The manipulation of polynomials is essential in
numerous applications of linear algebra and symbolic computation.
Partial Derivatives in Arithmetic Complexity and Beyond is devoted
mainly to the study of polynomials from a computational
perspective. It illustrates that one can learn a great deal about
the structure and complexity of polynomials by studying (some of)
their partial derivatives. It also shows that partial derivatives
provide essential ingredients in proving both upper and lower
bounds for computing polynomials by a variety of natural arithmetic
models. It goes on to look at applications which go beyond
computational complexity, where partial derivatives provide a
wealth of structural information about polynomials (including their
number of roots, reducibility and internal symmetries), and help us
solve various number theoretic, geometric, and combinatorial
problems. Partial Derivatives in Arithmetic Complexity and Beyond
is an invaluable reference for anyone with an interest in
polynomials. Many of the chapters in these three parts can be read
independently. For the few which need background from previous
chapters, this is specified in the chapter abstract.
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