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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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Reference request: ordered list of dimensions of components of a variety?

You can prove this without needing to keep track of dimensions. Suppose that $W$ and $V$ are real algebraic sets with $W \subsetneq V$, and that there exists a surjective polynomial map $\phi: W \twoh …
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