Algebraic geometry: Difference between revisions
Jump to navigation
Jump to search
imported>Holger Kley (A few introductory sentences; perhaps the start of a project.) |
imported>Holger Kley No edit summary |
||
| Line 1: | Line 1: | ||
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 | 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.'' | ||
--[[User:Holger Kley|Holger Kley]] 13:23, 6 December 2007 (CST) | --[[User:Holger Kley|Holger Kley]] 13:23, 6 December 2007 (CST) | ||
Revision as of 13:24, 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 would be defined by polynomials. --Holger Kley 13:23, 6 December 2007 (CST)
