Algebraic geometry: Difference between revisions
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imported>Holger Kley (New page: As the name suggests, algebraic geometry is the study of geometric objects defined by algebraic equations. For example, a parabola, such as all solutions <math>(x,y)</math> of the equatio...) |
imported>Holger Kley (A few introductory sentences; perhaps the start of a project.) |
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As the name suggests, algebraic geometry is the study of geometric objects defined by algebraic equations. For example, a parabola, such as all solutions <math>(x,y)</math> of the equation <math>y - x^2 = 0</math>, is one such object, whereas the graph of the exponential function---all solutions <math>(x,y)</math> of the equation <math>y - e^x = 0</math>---is not. The key distinction is that the equation defining the first example is a polynomial equation, whereas the second cannot be represented by polynomial equations, even implicitly. | As the name suggests, algebraic geometry is the study of geometric objects defined by algebraic equations. For example, a parabola, such as all solutions <math>(x,y)</math> of the equation <math>y - x^2 = 0</math>, is one such object, whereas the graph of the exponential function---all solutions <math>(x,y)</math> of the equation <math>y - e^x = 0</math>---is not. The key distinction is that the equation defining the first example is a polynomial equation, whereas the second cannot be represented by polynomial equations, even implicitly. In fact, in the present context, a reasonable and useful first approximation of the adjective ``algebraic'' is ''defined by polynomials.'' | ||
--[[User:Holger Kley|Holger Kley]] 13:23, 6 December 2007 (CST) | |||
Revision as of 14:23, 6 December 2007
As the name suggests, algebraic geometry is the study of geometric objects defined by algebraic equations. For example, a parabola, such as all solutions of the equation , is one such object, whereas the graph of the exponential function---all solutions of the equation ---is not. The key distinction is that the equation defining the first example is a polynomial equation, whereas the second cannot be represented by polynomial equations, even implicitly. In fact, in the present context, a reasonable and useful first approximation of the adjective ``algebraic is defined by polynomials. --Holger Kley 13:23, 6 December 2007 (CST)
