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Is the study of algebraic curve is techniquely equal to the advanced division of analytic geometry, if not, what is the difference?

And what is other branch of advanced analytic geometry called?

in addition, what is the differece between algebraic geometry and algebraic curves?

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In algebraic geometry one studies algebraic varieties, which locally look like the zero locus of some polynomials. A curve is a one-dimensional variety.

I don't hear people use the term "analytic geometry" much these days, but it could mean the study of analytic spaces, i.e. geometric objects which locally look like the the zero locus of some real or complex analytic functions. There is some overlap with algebraic geometry, but neither is a special case of the other.

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The "analytic geometry" I am accustomed to amounts to using coordinates to solve problems that were usually done by synthetic (i.e. in the manner of Euclid and cohorts) means. –  J. M. May 6 '12 at 17:40

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