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The fundamental objects of study in algebraic geometry are algebraic Varieties , geometric manifestations of Solutions of systems of Polynomial Equations . Plane Algebraic Curve s, which include Lines , Circle s, Parabola s, Lemniscate s, and Cassini Oval s, form one of the best studied classes of algebraic varieties. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve relative position of different curves and relations between the curves given by different equations. Descartes 's idea of Coordinates is central to algebraic geometry, but it has undergone a series of remarkable transformations beginning in the early 19th century. Before then, the coordinates were assumed to be tuples of Real Number s, but first Complex Number s, and then elements of an arbitrary Field became acceptable. Homogeneous Coordinates of Projective Geometry offered an extension of the notion of coordinate system in different direction, and enriched the scope of algebraic geometry. Much of the development of algebraic geometry in the 20th century occurred within abstract algebraic framework, with increasing emphasis being placed on 'intrinsic' properties of algebraic varieties not dependent on any particular way of embedding the variety in an ambient coordinate space; this parallels developments in Topology and Complex Geometry . One key distinction between classical projective geometry of 19th century and modern algebraic geometry, in the form given to it by Grothendieck and Serre , is that the former is concerned with the more geometric notion of a point, while the latter emphasizes the more analytic concepts of a Regular Function and a Regular Map and extensively draws on Sheaf Theory . Another important difference lies in the scope of the subject. Grothendieck's idea of scheme provides the language and the tools for geometric treatment of arbitrary Commutative Ring s and, in particular, bridges algebraic geometry with Algebraic Number Theory . Andrew Wiles 's celebrated Proof of Fermat's Last Theorem is a vivid testament to the power of this approach. André Weil , Grothendieck, and Deligne also Demonstrated that the fundamental ideas of topology of Manifolds have deep analogues in algebraic geometry over Finite Field s. ZEROS 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 two-dimensional Sphere in three-dimensional Euclidean Space R3 could be defined as the set of all points (''x'',''y'',''z'') with : A "slanted" circle in R3 can be defined as the set of all points (''x'',''y'',''z'') 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 C, but many of the same results are true if we assume only that ''k'' is Algebraically Closed . We define '''A'''n(''k'') (or more simply '''A'''''n'', when ''k'' is clear from the context), called the '''affine n-space over k''', to be ''k''''n''. The purpose of this apparently superfluous notation is to emphasize that one `forgets' the vector space structure that ''k''n carries. Abstractly speaking, '''A'''''n'' is, for the moment, just a collection of points. A function ''f'' : A''n'' → A1 is said to be '''regular''' if it can be written as a polynomial, that is, if there is a polynomial ''p'' in ''k'' {Link without Title} such that ''f''(''t''1,...,''t''''n'') = ''p''(''t''1,...,''t''''n'') for every point (''t''1,...,''t''''n'') of A''n''. 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 A''n'' as ''k'' A''n''" class="copylinks" target="_blank">{Link without Title} . 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 ''k'' An" class="copylinks" target="_blank">{Link without Title} . The ''vanishing set of S'' (or ''vanishing locus'') is the set ''V''(''S'') of all points in A''n'' where every polynomial in ''S'' vanishes. In other words, |
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