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Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problematics of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory. Initially a study of polynomial equations in many variables, the subject of algebraic geometry starts where equation solving leaves off, and it becomes at least as important to understand the totality of solutions of a system of equations, as to find some solution; this does lead into some of the deepest waters in the whole of mathematics, both conceptually and in terms of technique. The fundamental objects of study in algebraic geometry are algebraic varieties, geometric manifestations of solutions of systems of polynomial equations. Plane algebraic curves, which include lines, circles, parabolas, lemniscates, and Cassini ovals, 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. 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 rings 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 fields. In classical algebraic geometry, the main objects of interest are the vanishing sets of collections of polynomials, 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
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