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Semialgebraic geometry is essentially real algebraic geometry but with the defining polynomial relations allowed to be inequalities rather than just equalities.

This doesn't make sense over $\mathbb{C}$ because it isn't an ordered field, but we do have something not present over $\mathbb{R}$ : a nontrivial automorphism, namely complex conjugation.

So, instead of replacing polynomial equations with polynomial inequalities we can instead replace those polynomials in $z_i$ with polynomial in $z_i$ and $\bar{z_i}$

For example, we could have the equation $z = \bar{z}$, which defines the real line.

Is there a name for this theory, the study of graphs of polynomials in complex variables and their conjugates? Note, I am not looking for Hodge theory, which starts with a complex variety and then looks at differential forms in conjugate variables -- I want to start with a generalised variety that uses conjugate variables in its definition, as in the above examples of the real line.

J. W. Tanner
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Harry Wilson
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    Writing $z = x + iy$ with $x$ and $y$ real, the sets ${x, y}$ and ${z, \bar{z}}$ generate the same complex vector space. If I understand what you're asking, it therefore appears this set-up with $n$ complex variables amounts to real algebraic geometry in $2n$ variables. – Andrew D. Hwang Feb 19 '24 at 16:36

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