The Set of Separable States has no Finite Semidefinite Representation Except in Dimension $$3\times 2$$

Given integers $$n \ge m$$ n ≥ m , let $$\text {Sep}(n,m)$$ Sep ( n , m ) be the set of separable states on the Hilbert space $$\mathbb {C}^n \otimes \mathbb {C}^m$$ C n ⊗ C m . It is well-known that for $$(n,m)=(3,2)$$ ( n , m ) = ( 3 , 2 ) the set of separable states has a simple description using semidefinite programming: it is given by the set of states that have a positive partial transpose. In this paper we show that for larger values of n and m the set $$\text {Sep}(n,m)$$ Sep ( n , m ) has no semidefinite programming description of finite size. As $$\text {Sep}(n,m)$$ Sep ( n , m ) is a semialgebraic set this provides a new counterexample to the Helton–Nie conjecture, which was recently disproved by Scheiderer in a breakthrough result. Compared to Scheiderer’s approach, our proof is elementary and relies only on basic results about semialgebraic sets and functions.

Counting algebraic points in expansions of o-minimal structures by a dense set

The Pila–Wilkie theorem states that if a set $X\subseteq \mathbb{R}^n$ is definable in an o-minimal structure $\mathcal{R}$ and contains ‘many’ rational points, then it contains an infinite semialgebraic set. In this paper, we extend this theorem to an expansion $\widetilde{\mathcal{R}}=\langle {\mathcal{R}}, P\rangle$ of ${\mathcal{R}}$ by a dense set P, which is either an elementary substructure of ${\mathcal{R}}$, or it is $\mathrm{dcl}$-independent, as follows. If X is definable in $\widetilde{\mathcal{R}}$ and contains many rational points, then it is dense in an infinite semialgebraic set. Moreover, it contains an infinite set which is ${\emptyset}$-definable in $\langle \overline{\mathbb{R}}, P\rangle$, where $\overline{\mathbb{R}}$ is the real field. Along the way we introduce the notion of the ‘algebraic trace part’ $X^{{\, alg}}_t$ of any set $X\subseteq \mathbb{R}^n$, and we show that if X is definable in an o-minimal structure, then $X^{{\, alg}}_t$ coincides with the usual algebraic part of X.

Real Tropicalization and Analytification of Semialgebraic Sets

Let $K$ be a real closed field with a nontrivial non-archimedean absolute value. We study a refined version of the tropicalization map, which we call real tropicalization map, that takes into account the signs on $K$. We study images of semialgebraic subsets of $K^n$ under this map from a general point of view. For a semialgebraic set $S \subseteq K^n$ we define a space $S_r^{{\operatorname{an}}}$ called the real analytification, which we show to be homeomorphic to the inverse limit of all real tropicalizations of $S$. We prove a real analogue of the tropical fundamental theorem and show that the tropicalization of any semialgebraic set is described by tropicalization of finitely many inequalities, which are valid on the semialgebraic set. We also study the topological properties of real analytification and tropicalization. If $X$ is an algebraic variety, we show that $X_r^{{\operatorname{an}}}$ can be canonically embedded into the real spectrum $X_r$ of $X$, and we study its relation with the Berkovich analytification of $X$.

ON THE SUBSTITUTION THEOREM FOR RINGS OF SEMIALGEBRAIC FUNCTIONS

Let$R\subset F$be an extension of real closed fields, and let${\mathcal{S}}(M,R)$be the ring of (continuous) semialgebraic functions on a semialgebraic set$M\subset R^{n}$. We prove that every$R$-homomorphism${\it\varphi}:{\mathcal{S}}(M,R)\rightarrow F$is essentially the evaluation homomorphism at a certain point$p\in F^{n}$adjacent to the extended semialgebraic set$M_{F}$. This type of result is commonly known in real algebra as a substitution lemma. In the case when$M$is locally closed, the results are neat, while the non-locally closed case requires a more subtle approach and some constructions (weak continuous extension theorem, appropriate immersion of semialgebraic sets) that have interest of their own. We consider the same problem for the ring of bounded (continuous) semialgebraic functions, getting results of a different nature.