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ZEROES OF SIMULTANEOUS POLYNOMIALS In classical algebraic geometry, the main objects of interest are the vanishing sets of collections of Polynomial s, meaning the set of all points that simultaneously satisfy one or more polynomial equations. For instance, the three-dimensional Sphere in three-dimensional Euclidean Space could be defined as the set of all points with :. A "slanted" circle in can be defined as the set of all points which satisfy the two polynomial equations :, :. AFFINE VARIETIES First we start with a Field ''k''. In classical algebraic geometry, this field was always , the complex numbers, but many of the same results are true if we assume only that ''k'' is Algebraically Closed . We define , called the affine n-space over k, to be ''kn''. The purpose of this apparently superfluous notation is to emphasize that one `forgets' the vector space structure that ''kn'' carries. Abstractly speaking, is, for the moment, just a collection of points. Henceforth we will drop the ''k'' in and instead write . Define a function : to be regular if it can be written as a polynomial, that is, if there is a polynomial ''p'' in k such that for each point :(''t''1,...,''t''''n'') of , f Regular functions on affine ''n''-space are thus exactly the same as polynomials over ''k'' in ''n'' variables. We will write the regular functions on as . We say that a polynomial ''vanishes'' at a point if evaluating it at that point gives zero. Let ''S'' be a set of polynomials in . The ''vanishing set of S'' (or ''vanishing locus'') is the set ''V''(''S'') of all points in where every polynomial in ''S'' vanishes. In other words, |
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