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Real algebraic geometry is the study of real solutions to algebraic equations with real coefficients. Its methods are rather different from classical algebraic geometry, which is typically done over an algebraically closed field (like the complex numbers).

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Real algebraic geometry vs. algebraic geometry

For an easy introduction to RAG, you could read van den Dries book "Tame topology and o-minimal structures": he treats the more general notion of o-minimal structures instead of real closed fields, an …
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