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Fearless Symmetry by Robert Gross, Avner Ash

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Wenowcomefacetofacewiththemotivatingproblemof
this book: equations and how to solve them. Although as
number theorists we are primarily interested in integral
or fractional solutions of systems of polynomial equations
with integral coefficients (“Diophantine equations”), it
will be helpful to consider also
equations and their solutions in general, and
solutions to equations involving number systems other
than the integers.
In general, we cannot simply list all of the s olutions
of a given system of equations. Often, we cannot find all
of them, or there are infinitely many solutions, or there
are no solutions—but we may not know that (yet). Even if
we can list all the solutions to some system of equations,
such a list does not give us full understanding of the
structure of the set of solutions. In this chapter we broach
the question: “What are interesting structural properties
of sets of solutions?”
Also in this chapter, we define variety, a concept that
gives a concise way to discuss solutions in varying number
systems to a fixed system of polynomial equations.

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