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I am a Engineering postgraduate student and want to learn algebraic geometry especially to count the dimension of the solution space of system of polynomial equations. Can you recommend a book and if possible a video lecture that one can learn the fundamentals. My knowledge in linear algebra is very good but algebraic geometry is almost zero.

Is differential geometry something related to algebraic geometry?

There are some links stated in http://mathoverflow.net/questions/78037?sort=votes#sort-top but i am not sure which one would be easy to understand for engineering students.

Thank you in advance.

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The books by Cox, Little, and O'Shea are focused on computations. Using Algebraic Geometry seems like a promising title, for example. –  Dylan Moreland May 15 '12 at 19:53

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I would recommend Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra by Cox, Little and O'Shea if your initial algebraic geometry knowledge is almost zero. It is written for advanced undergraduate students in mathematics so it doesn't assume prior exposure to any of the abstract structures (groups, rings, etc.) prevalent in the topic. Also, it puts a very strong emphasis on the computational aspects of the subject (no category theory!) and $actually$ explains everything so it seems perfect for you. The many pictures and the great writing style make for a refreshing and enjoyable read. It even has a section on applications to robotics!

Differential geometry and algebraic geometry may study similar objects at times but are completely different approaches (and algebraic geometry is ${much}$ more general). Differential geometry heavily relies on generalizations of the techniques of calculus, applying them to manifolds (smooth objects) equipped with a differential structure. On the other hand, algebraic geometry studies the set of common zeroes of families of polynomials using much more algebraic techniques. These common zeroes of families of polynomials may have many "kinks" causing differential techniques to fail.

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Thank you. Can you suggest a video lecture? –  karthik May 16 '12 at 4:37

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