Algebraic geometry: Difference between revisions
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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. In fact, in the present context, a reasonable and useful first approximation of the adjective ''algebraic'' would be ''defined by polynomials.'' | 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'' would be ''defined by polynomials.'' | ||
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Revision as of 02:38, 7 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 would be defined by polynomials.
