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	Algebraic geometry - Wikipedia, the free encyclopedia
		Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry.
		Algebraic geometry was developed largely by the Italian geometers in the early part of the 20th century.
		Commutative algebra (earlier known as elimination theory and then ideal theory, and refounded as the study of commutative rings and their modules) had been and was being developed by David Hilbert, Max Noether, Emanuel Lasker, Emmy Noether, Wolfgang Krull, and others.
	en.wikipedia.org /wiki/Algebraic_geometry (1802 words)

	PlanetMath: the arithmetic of elliptic curves (Site not responding. Last check: 2007-11-07)
		The theory of elliptic curves is a very rich mix of algebraic geometry and number theory (arithmetic geometry).
		The conductor of an elliptic curve is an integer quantity that measures the arithmetic complexity of the curve (the entry contains examples).
		This is version 9 of the arithmetic of elliptic curves, born on 2005-03-01, modified 2005-03-01.
	planetmath.org /encyclopedia/ArithmeticOfEllipticCurves.html (547 words)

	Curve definitions
		Algebraic curve : A curve whose cartesian equation can be expressed in terms of powers of x and y together with the operations of addition, subtraction, multiplication and division.
		Pedal curve : Given a curve C then the pedal curve of C with respect to a fixed point O (called the pedal point) is the locus of the point P of intersection of the perpendicular from O to a tangent to C.
		Transcendental curve : A curve of the form f(x,y) = 0 where f(x,y) is not a polynomial in x and y.
	www-groups.dcs.st-andrews.ac.uk /%7Ehistory/Curves/Definitions2.html (1325 words)

	8.1 Algebraic Curves
		Curves that can be given in implicit form as f(x,y)=0, where f is a polynomial, are called algebraic.
		Thus conics (Section 7) are algebraic curves of degree two.
		Curves of degree three already have a great variety of shapes, and only a few common ones will be given here.
	www.geom.uiuc.edu /docs/reference/CRC-formulas/node33.html (846 words)

	Xah: Special Plane Curves: Naming and Classification of Curves
		To determine whether a curve is algebraic requires graduate level math knowledge, and is beyond this project's scope.
		All the curves covered here are such that when you keep magnifying parts of the curve, it'll eventually looks like a line, unless you are magnifying a cusp point.
		Isoptic of a given curve C and a given angle α is the locus of a point P such that P is the intersection of tangents of C that meets in angle α.
	www.xahlee.org /SpecialPlaneCurves_dir/Intro_dir/familyIndex.html (1043 words)

	Algebraic Curves
		The genus of an algebraic curve equals (d-1)(d-2)/2 minus the sum of the delta invariants.
		This is an algebraic extension of C(x) of degree degree(f,y).In some applications (integration of algebraic functions, and the method that algcurves[parametrization] uses) one needs to be able to recognize the poles of elements in the function field.
		An algebraic curve is normally not viewed as lying in the affine plane C^2 (where C is the field of constants) but in the projective plane P^2(C).
	www.math.fsu.edu /~hoeij/algcurves.html (2858 words)

	MA4121 Projective Curves
		Curves in the plane like lines, circles, ellipse, parabola and hyperbola can be described by polynomial equations, the first by a linear equation the other by quadratic equations.
		This module aims to introduce basic ideas of Algebraic Geometry, to show how basic ideas from pure mathematics could be brought together in one of the very beautiful subjects of mathematics and to demonstrate the power of the interaction between algebra and geometry.
		Affine curves, homogeneous coordinates, projective plane, varieties, irreducible components, coordinate transformations, minimal polynomials, projective and plane algebraic curves, intersection of curves, singular and simple points, the degree of a curve, Bezout's Theorem, points of inflection and cubics.
	www.mcs.le.ac.uk /Modules/MA-02-03/MA4121.html (511 words)

	Research in Algebraic Geometry
		The study of curves is the most classical part of algebraic geometry.
		Much of the work in the field of algebraic cycles is organized around three major conjectures: the Hodge conjecture, the Tate conjecture and the generalized Birch-Swinnerton Dyer conjecture.
		Algebraic Geometry and theoretical physics have found common interest in the study of "Calabi-Yau Manifolds".
	www.math.duke.edu /%7Eschoen/researchalggeo.html (712 words)

	Algebraic Curves (B9) (Site not responding. Last check: 2007-11-07)
		Algebraic curves appear in most areas of mathematics,from string theory to number theory, from differential equations to coding theory.
		In this course however, we adopt the algebraic approach (essentially, an algebraic curve is given by a polynomial in two variables), working over an arbitrary algebraically closed field, but assuming characteristic zero on occasions to simplify things.
		Most of the standard ideas and tools for studying algebraic curves will be introduced, including the central concept of the genus of a curve.
	www.maths.cam.ac.uk /undergrad/courseinfo/coursesII/text/node45.html (248 words)

	ACA 2003 Session T1: Computational aspects of algebraic curves
		Algebraic curves have been studied for a long time, however there are still many problems left unanswered some of which with a long history.
		Specifically, given an algebraic variety $X/R$ and a differential operator $\delta :R \to R$, it is possible to construct a prolongation sequence of varieties $.
		Determining the automorphism group Aut (X) of this curve, the field of moduli, and the minimal field of definition are some of the problems of classical algebraic geometry.
	www.math.unm.edu /ACA/2003/Sessions/T1.html (1376 words)

	Vertex Algebras and Algebraic Curves (Site not responding. Last check: 2007-11-07)
		This book is an introduction to the theory of vertex algebras with a particular emphasis on the relationship between vertex algebras and the geometry of algebraic curves.
		The notion of a vertex algebra is introduced in the book in a coordinate-independent way, allowing the authors to give global geometric meaning to vertex operators on arbitrary smooth algebraic curves, possibly equipped with some additional data.
		From this perspective, vertex algebras appear as the algebraic objects that encode the geometric structure of various moduli spaces associated with algebraic curves.
	www.math.berkeley.edu /~frenkel/BOOK (494 words)

	curves.html (Site not responding. Last check: 2007-11-07)
		From that time, a plane curve could be defined as the graph of an equation in two variables, and a space curve as the intersection of the graphs of two equations in three variables.
		The set of all points traced out as P moves allway staying r units from C (shortest distance) is called a parallel curve to C. The parallell curves to a line are lines; the parallel curves to a circle are circles.
		As in Bolzano's case, Hilbert's space filling curve is the uniform limit of a sequence of curves.
	www.ms.uky.edu /~carl/ma330/html/curves1.html (611 words)

	Untitled Document
		My description: We encountered algebraic curves long before college; for example we are all familiar with the graphs (or ``zero loci'') of y=x^2, x^2+y^2=1 etc. in R^2.
		Amazing fact: there is a natural correspondence between these surfaces and the plane curves we introduced above, which makes much of the study assailable from the (sometimes easier) points of view of algebra and projective geometry.
		Algebraic curves: Zero sets of polynomials in two variables defined over Q, Generalization to curves in CP^2.
	www.math.uga.edu /undergraduate/MATH_4300_6300_rulla_0803.html (822 words)

	CMJ Contents: March 1998
		By constructing a mathematical model for the motion, which involves finding and applying the moment of inertia of a ball, a convincing case is made that Galileo's data are the record of a genuine experiment, and not the result of a thought experiment as some historians had once maintained.
		Here an alternative approach is given, based on the geometrically appealing axiom that of two continuously differentiable functions, one of whose derivative is larger in absolute value than that of the other throughout an interval, the graph of the one with the greater derivative has the greater arc length.
		Three cases are considered: the buckled shape is an isosceles triangle, an arc of a circle, or one arch of a sinusoidal curve.
	www.maa.org /pubs/cmj_mar98.html (1134 words)

	Forward Iterates of Algebraic Curves under a Holomorphic self map of the Projective Plane. (Site not responding. Last check: 2007-11-07)
		The sequence of curves in the first column are the iterates of f applied to the curve x=0.
		The sequence of curves in the second column are the iterates of f applied to the curve y=0.
		The sequence of curves in the third column are the iterates of f applied to the curve z=0.
	www.math.wichita.edu /~robertson/forimages (100 words)

	Convexity Preserving Interpolation by Algebraic Curves and Surfaces -- from Mathematica Information Center
		The problem of interpolation by a convex curve to the vertices of a convex polygon is considered.
		This is extended to a solution of a general Hermite-type problem, in which the curve also interpolates to one or two prescribed tangents at any desired vertices of the polygon.
		Several properties of this family of algebraic curves are discussed.
	library.wolfram.com /infocenter/Articles/3287 (127 words)

	Xah: Special Plane Curves: References
		It proceeds through the standard properties of parametric curves, the classification of limacons, and an account of envelopes of curve families, and finally to projective curves, their relationship to algebraic curves, and their application to asymptotes and boundedness.
		Here's the description from the back cover: This is a genuine introduction to plane algebraic curves from a geometric viewpoint, designed as a first text for undergraduates in mathematics, or for postgraduate and research workers in the engineering and physical sciences.
		A minimal amount of algebra leads to the famous theorem of Bezout, whilst the ideas of linear systems are used to discuss the classical group structure on the cubic.
	www.xahlee.org /SpecialPlaneCurves_dir/Intro_dir/references.html (2632 words)

	Visualization of some simple algebro-geometric ideas
		In the first elementary study of complex plane algebraic curves, appearance of the real part of the curve may be misleading for students who begin to study the subject.
		If it has one double and one simple root, the curve is an "alpha" curve with one double point of self-intersection with different tangent lines.
		Topology of the singular point of the "alpha" curve is clearly visible in this picture: its small neighbourhood consists of two disks, glued in one point (a bouquet of two disks).
	members.tripod.com /vismath2/lip (879 words)

	14: Algebraic geometry (Site not responding. Last check: 2007-11-07)
		Algebraic geometry combines the algebraic with the geometric for the benefit of both.
		Singular Package: Singular is a computer algebra system for singularity theory and algebraic geometry developed by G.-M. Greuel, G. Pfister, H.
		Note that many computations in algebraic geometry are really computations in polynomials rings, hence computational commutative algebra applies.
	www.math.niu.edu /~rusin/known-math/index/14-XX.html (523 words)

	Elliptic curves (Site not responding. Last check: 2007-11-07)
		On the conjecture of Birch and Swinnerton-Dyer for elliptic curves, by Cristian D. Gonzalez-Aviles.
		The average rank of an algebraic family of elliptic curves, J.
		Beppo Levi and the arithmetic of elliptic curves, by Norbert Schappacher.
	www.fermigier.com /fermigier/elliptic.html.en (746 words)

	The Math Forum - Math Library - Algebraic Geom. (Site not responding. Last check: 2007-11-07)
		Algebraic Geometry preprints, from the U.C. Davis front end for the xxx.lanl.gov e-Print archive, a major site for mathematics preprints that has incorporated many formerly independent specialist archives.
		An area of algebraic geometry that deals with nonsingular curves of genus 1 - in English, solutions to equations y^2 = x^3 + A x + B. It has important connections to number theory and in particular to factorization of ordinary integers (and thus to cryptography).
		An introduction to computational algebraic geometry and commutative algebra at the undergraduate level, with discussions of systems of polynomial equations ("ideals"), their solutions ("varieties"), and how these objects can be manipulated ("algorithms").
	mathforum.org /library/topics/algebraic_g (1954 words)

	PlanetMath: Riemann-Roch theorem for curves (Site not responding. Last check: 2007-11-07)
		be a projective nonsingular curve over an algebraically closed field.
		Cross-references: canonical, genus, divisor, field, algebraically closed, curve, nonsingular
		This is version 7 of Riemann-Roch theorem for curves, born on 2001-12-12, modified 2005-03-18.
	planetmath.org /encyclopedia/RiemannRochTheorem.html (70 words)

	Algebraic Geometry (Site not responding. Last check: 2007-11-07)
		Algebraic Geometry is a subject with historical roots in analytic geometry.
		It subsumes most of commutative algebra and much of algebraic number theory, and overlaps with differential geometry, modern "analytic geometry" (complex manifolds), Lie groups, representation theory, theoretical physics, and to a lesser extent the theory of partial differential equations.
		In addition to being one of the central disciplines of pure mathematics, algebraic geometry has developed an applied side which is linked to problems in computational complexity and the theory of algorithms, symbolic computation, robotics, control theory, computational geometry, geometric modeling, image recognition, computer vision, and scientific visualization.
	www.math.tamu.edu /~Peter.Stiller/agpage.html (224 words)

	14H52: Elliptic Curves (Site not responding. Last check: 2007-11-07)
		This is a fascinating area of algebraic geometry dealing with nonsingular curves of genus 1 -- in English, solutions to equations y^2 = x^3 + A x + B. It turns out to have important connections to number theory and in particular to factorization of ordinary integers (and thus to cryptography).
		Elliptic curves also played a role in the recent resolution of the conjecture known as Fermat's Last Theorem.
		Two (unstructured) equations equations in three unknowns lead to an elliptic curve (although integer points are not fully known).
	www.math.niu.edu /~rusin/known-math/index/14H52.html (750 words)

	Things of interest to number theorists
		The Selmer group is for an isogeny, over a number field, from an abelian variety to the Jacobian of a curve where the kernel of the isogeny is killed by a power of a prime.
		In this paper we prove that p-Selmer groups for elliptic curves can be arbitrarily large if one ranges over number fields of degree at most g+1 over the rationals where g is the genus of X_0(p).
		This is a group variety whose dimension is the same as the genus of the curve and can also be considered to be the divisor classes of degree 0 of the curve.
	math.scu.edu /~eschaefe/nt.html (1525 words)

	Algebraic Curves That Work Better (Site not responding. Last check: 2007-11-07)
		An algebraic curve is defined as the zero set of a polynomial in two variables.
		Algebraic curves are practical for modeling shapes much more complicated than conics or superquadrics.
		The main drawback in representing shapes by algebraic curves has been the lack of repeatability in fitting algebraic curves to data.
	www.lems.brown.edu /~jpt/cvpr99.html (183 words)

	AG+CA
		Algebraic Geometry is one of the oldest of the classical mathematical disciplines.
		The development of algebraic geometry throughout the past centuries occurred in waves and was influenced by different schools, each of them using a different language.
		The main task of algebraic geometry, the investigation of the global and the local structure of varieties brought forth the discipline of singularity theory.
	www.mathematik.uni-kl.de /mathint/agca/alg_geo.htm (1371 words)

	Algebraic Geometry and Arithmetic Curves (Site not responding. Last check: 2007-11-07)
		This book provides a general introduction to the theory of schemes, followed by applications to arithmetic surfaces and to the theory of reduction of algebraic curves.
		Another feature of this highly valuable book on algebraic and arithmetic geometry is provided by the vast amount of illustrating, theoretically important examples as well as by the approximately six hundred included exercises.
		As stated before, this book is unique in the current literature on algebraic and arithmetic geometry, therefore a highly welcome addition to it, and particularly suitable for readers who want to approach more specialized works in this field with more ease.
	www.math.u-bordeaux.fr /~liu/Book/index.html (281 words)

	General Algebraic Curves
		Its main features are flexible tools for translating between affine and projective curves, the calculation of geometric genus of any plane curve and the explicit manipulation of divisors on curves.
		In particular, the functions and usage for different kinds of curves is converging to a standard.
		Most of the new curve functionality is based on our new function field machinery, but in the context of curves the results are available directly without function field knowledge.
	magma.maths.usyd.edu.au /magma/Features/node209.html (116 words)

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