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  • $\begingroup$ Thank you very much for your answer! Perhaps I should have been more clear: by "modern algebraic geometry", I refer to the so-called "Grothendieck and Serre language of algebraic geometry". Could you please be more specific about the "homotopy methods" and "numerical algebraic geometry"? I appreciate the reference to the paper, but it would be nice if you could state some context and explanation regarding how the methods are modern but concrete. (For example, a rough overview, say one or two lines, regarding the context of the subject in mathematics.) $\endgroup$ Commented Jul 13, 2010 at 13:47
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    $\begingroup$ "Modern" as in the methods were developed very recently (last 15 years) rather than from Grothendieck, Serre onwards. Concrete as in computational, algorithmic and calculating solutions of specific systems of polynomials rather than abstract theory about phenomena that can occur. For some context, see Google books for preface of "Interactions of classical and numerical algebraic geometry". See also wikipedia article on homotopy. $\endgroup$ Commented Jul 13, 2010 at 18:39
  • $\begingroup$ Also pages 16-19 of science.nd.edu/documents/publications/renaissance.pdf $\endgroup$ Commented Jul 13, 2010 at 18:47
  • $\begingroup$ Thank you for the clarification. I do know what homotopy means (in the context of algebraic topology), but I was not so sure how the "methods" of homotopy applied in numerical algebraic geometry, hence my comment. But I will take a look at those papers. (When I think of homotopy and algebraic geometry, I think "Hodge theory", but I guess that is not what numerical algebraic geometry is about.) $\endgroup$ Commented Jul 14, 2010 at 4:53
  • $\begingroup$ Hi, Maclean. Which books should i have to read to start to learn Numerical algebraic geometry? I have already self-studied manifold, functional analysis and commutative algebra and will study Introduction to Complex Analysis in Several Variables and algebraic curves by Fulton $\endgroup$ Commented Aug 17, 2019 at 10:02