**Definition.** A subset $S$ of $\mathbb{R}^n$ is called *semi-algebraic* if $S$ is a finite Boolean combination of sets of the form $\{ x \in \mathbb{R}^n \mid p(x) \ge 0\}$, where $p \in \mathbb{R}[x]$.

View $\mathbb{C}^n$ as a real $2n$-dimensional vector space and let $\mathsf{T}: \mathbb{C}^n \longrightarrow \mathbb{R}^{2n}$ be the bijective linear transformation such that $$ z \in \mathbb{C}^n \stackrel{\mathsf{T}}{\longmapsto} \begin{bmatrix} \Re z_1 \\ \Im z_1 \\ \vdots \\ \Re z_n \\ \Im z_n \end{bmatrix} \in \mathbb{R}^{2n}. $$

As suggested in a comment here, call $S \subseteq \mathbb{C}^n$ *complex semi-algebraic* or *semi-algebraic in $\mathbb{C}^n$* if $\mathsf{T}(S)$ is semi-algebraic in $\mathbb{R}^{2n}$.

**Question.** Is there an equivalent definition that involves, perhaps, a set of inequalities involving the modulus?

As an example, if $S := \{z \in \mathbb{C} \mid 1 - \vert z^2 \vert \ge 0\}$, then $\mathsf{T}(S) = \{ x \in \mathbb{R}^2 \mid 1 - (x^2 + y^2) \ge 0 \}$ is semi-algebraic in $\mathbb{R}^2$.

**Edit:** I've decided to add some context given that someone decided to downvote this post.

The *nonnegative inverse eigenvalue problem* (NIEP) is to characterize the spectra of (entrywise) nonnegative matrices. The NIEP is unsolved for matrices of order greater than four.

Let $x \in \mathbb{C}^n$. Say $x$ is *realizable* if the multiset $\Lambda(x) := \{x_1,\dots,x_n\}$ is the spectrum of a nonnegative matrix of order $n$. Let $\mathbb{L}^n := \{ x \in \mathbb{C}^n \mid \Lambda(x) = \sigma(A),~A \in \mathsf{M}_n(\mathbb{R}),~A \ge 0 \}$. (There are some well-known necessary conditions on this set, but we do not mention them here.)

In reviewing the NIEP, Bharali and Holtz [MR2399570] state

Finally, it follows from the Tarski–Seidenberg theorem [38, 29] that all realizable $n$-tuples form a

semialgebraic set(see also [16]); i.e., for any given $n$,there exist only finitely many polynomial inequalities that are necessary and sufficient for an $n$-tuple $\Lambda$ to be realizable[emphasis added] as the spectrum of some nonnegative matrix $A$ (this observation was communicated to us by Friedland).

As illustrated above, this is clearly not the case and demonstrates the need for defining a *complex semi-algebraic* set beyond what I proffered above. Furthermore, a rigorous, "local" definition would give us a certificate of what it means to *solve* the NIEP.

or morepolynomial equalities and inequalities. Are you restricting to semialgebraic sets of a special form? $\endgroup$ – Zach Teitler Aug 8 '19 at 2:1820more comments