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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