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The answer is indeed No. The most economic example up to now i think is mentioned Suvorov's. Independently examples was constructed by P. Mani, B. Jaggi, B. Sturmfels, N. White "Uniform oriented matroids without the isotopy property" (link to pdf)
My perhaps first by time example of simple line arrangement without isotopy property has 19 lines .

I will use language of line arrangements.

As @joseph-orourke correctly guessed in reply to @j.c. the stronger statement holds:

Any open basic semi-algebraic set (i.e the set defined by strict polynomial inequalities with integer coefficients) can be realized up to certain trivial stabilization by realization space of a simple pseudolines arrangement.

It was the initial target theorem due to some applications to Morse theory of vector functions. Reference is N. E. Mnev, "The universality theorems on the classification problem of configuration varieties and convex polytopes varieties" (link to pdf) Theorem A, statement 2). This old paper only briefly describes the content of unpublished phd (link to pdf) in Russian. Unfortunately this moment was not explicitly covered in later improvements and expositions of the proof. But it is really a simple addition to the main construction modernized by P.Shor "Stretchability of pseudolines is NP-hard" (link to pdf), H. Gunzel "The Universal Partition Theorem for Oriented Matroids" (link to pdf) and U. Richter-Gebert "Mne"v's Universality Theorem Revisited" (link to pdf). The point is that having as an input a system of strict algebraic inequalities the general machine produces “constructible” arrangement of pseudolines with realization space equivalent to the open semi-algebraic set defined by the system. “Constructible” means “coming from a free construction using ruler”. It has an order on the pseudolines such that every next is incident to not more then two points in general position defined as intersections of pervious pseudolines. Then such an arrangement can be perturbed to simple with the equivalent realization space by an inductive perurbation trick applied to pseudolines in the reverse order of the construction. The trick was introduced independently by B. Sturmfels and N. White "Constructing uniform oriented matroids without the isotopy property" (link to pdf), see also Shor Lemma 4 (link to pdf) and around.

The answer is indeed No. The most economic example up to now i think is mentioned Suvorov's. Independently examples was constructed by P. Mani, B. Jaggi, B. Sturmfels, N. White "Uniform oriented matroids without the isotopy property" (link to pdf)
My perhaps first by time example of simple line arrangement without isotopy property has 19 lines .

I will use language of line arrangements.

As @joseph-orourke correctly guessed in reply to @j.c. the stronger statement holds:

Any open basic semi-algebraic set (i.e the set defined by strict polynomial inequalities with integer coefficients) can be realized up to certain trivial stabilization by realization space of a simple pseudolines arrangement.

It was the initial target theorem due to some applications to Morse theory of vector functions. Reference is N. E. Mnev, "The universality theorems on the classification problem of configuration varieties and convex polytopes varieties" (link to pdf) Theorem A, statement 2). This old paper only briefly describes the content of unpublished phd (link to pdf) in Russian. Unfortunately this moment was not explicitly covered in later improvements and expositions of the proof. But it is really a simple addition to the main construction modernized by P.Shor "Stretchability of pseudolines is NP-hard" (link to pdf), H. Gunzel "The Universal Partition Theorem for Oriented Matroids" (link to pdf) and U. Richter-Gebert "Mne"v's Universality Theorem Revisited" (link to pdf). The point is that having as an input a system of strict algebraic inequalities the general machine produces “constructible” arrangement of pseudolines with realization space equivalent to the open semi-algebraic set defined by the system. “Constructible” means “coming from a free construction using ruler”. It has an order on the pseudolines such that every next is incident to not more then two points in general position defined as intersections of pervious pseudolines. Then such an arrangement can be perturbed to simple with the equivalent realization space by an inductive perurbation trick applied to pseudolines in the reverse order of the construction. The trick was introduced independently by B. Sturmfels and N. White "Constructing uniform oriented matroids without the isotopy property" (link to pdf), see also Shor Lemma 4 (link to pdf) and around.

For completeness i will add that the statement holds for uniform oriented matroids of rank >= 3 . It is not an immediate collary. Very nice reduction of any rank greater than 3 to rank 3 presented by @Arnau

The answer is indeed No. The most economic example up now i think is mentioned Suvorov's. Independently examples was constructed by P. Mani, B. Jaggi, B. Sturmfels, N. White "Uniform oriented matroids without the isotopy property"
My perhaps first by time example of simple line arrangement without isotopy property has 19 lines .

I will use language of line arrangements.

As @joseph-orourke correctly guessed in reply to @j.c. the stronger statement holds:

Any open basic semi-algebraic set (i.e the set defined by strict polynomial inequalities with integer coefficients) can be realized up to certain trivial stabilization by realization space of a simple pseudolines arrangement.

It was the initial target theorem due to some applications to Morse theory of vector functions. Reference is N. E. Mn"ev, The universality theorems on the classification problem of configuration varieties and convex polytopes varieties Theorem A, statement 2). This old paper only briefly describes the content of unpublished phd in Russian. Unfortunately this moment was not explicitly covered in later improvements and expositions of the proof. But it is really a simple addition to the main construction modernized by P.Shor, H. Gunzel and U. Richter-Gebert. The point is that having as an input a system of strict algebraic inequalities the general machine produces “constructible” arrangement of pseudolines with realization space equivalent to the open semi-algebraic set defined by the system. “Constructible” means “coming from a free construction using ruler”. It has an order on the pseudolines such that every next is incident to not more then two points in general position defined as intersections of pervious pseudolines. Then such an arrangement can be perturbed to simple with the equivalent realization space by an inductive perurbation trick applied to pseudolines in the reverse order of the construction. The trick was introduced independently by B. Sturmfels and N. White Constructing uniform oriented matroids without the isotopy property, see also Shor Lemma 4 and around.

The answer is indeed No. The most economic example up to now i think is mentioned Suvorov's. Independently examples was constructed by P. Mani, B. Jaggi, B. Sturmfels, N. White "Uniform oriented matroids without the isotopy property" (link to pdf)
My perhaps first by time example of simple line arrangement without isotopy property has 19 lines .

I will use language of line arrangements.

As @joseph-orourke correctly guessed in reply to @j.c. the stronger statement holds:

Any open basic semi-algebraic set (i.e the set defined by strict polynomial inequalities with integer coefficients) can be realized up to certain trivial stabilization by realization space of a simple pseudolines arrangement.

It was the initial target theorem due to some applications to Morse theory of vector functions. Reference is N. E. Mnev, "The universality theorems on the classification problem of configuration varieties and convex polytopes varieties" (link to pdf) Theorem A, statement 2). This old paper only briefly describes the content of unpublished phd (link to pdf) in Russian. Unfortunately this moment was not explicitly covered in later improvements and expositions of the proof. But it is really a simple addition to the main construction modernized by P.Shor "Stretchability of pseudolines is NP-hard" (link to pdf), H. Gunzel "The Universal Partition Theorem for Oriented Matroids" (link to pdf) and U. Richter-Gebert "Mne"v's Universality Theorem Revisited" (link to pdf). The point is that having as an input a system of strict algebraic inequalities the general machine produces “constructible” arrangement of pseudolines with realization space equivalent to the open semi-algebraic set defined by the system. “Constructible” means “coming from a free construction using ruler”. It has an order on the pseudolines such that every next is incident to not more then two points in general position defined as intersections of pervious pseudolines. Then such an arrangement can be perturbed to simple with the equivalent realization space by an inductive perurbation trick applied to pseudolines in the reverse order of the construction. The trick was introduced independently by B. Sturmfels and N. White "Constructing uniform oriented matroids without the isotopy property" (link to pdf), see also Shor Lemma 4 (link to pdf) and around.

The answer is indeed No. The most economic example up now i think is mentioned Suvorov's. Independently examples was constructed by P. Mani, B. Jaggi, B. Sturmfels, N. White "Uniform oriented matroids without the isotopy property"
My perhaps first by time example of simple line arrangement without isotopy property has 19 lines .

I will use language of line arrangements.

As @joseph-orourke correctly guessed in reply to @j.l. the stronger statement holds:

Any open basic semi-algebraic set (i.e the set defined by strict polynomial inequalities with integer coefficients) can be realized up to certain trivial stabilization by realization space of a simple pseudolines arrangement.

It was the initial target theorem due to some applications to Morse theory of vector functions. Reference is N. E. Mn"ev, The universality theorems on the classification problem of configuration varieties and convex polytopes varieties Theorem A, statement 2). This old paper only briefly describes the content of unpublished phd in Russian. Unfortunately this moment was not explicitly covered in later improvements and expositions of the proof. But it is really a simple addition to the main construction modernized by P.Shor, H. Gunzel and U. Richter-Gebert. The point is that having as an input a system of strict algebraic inequalities the general machine produces “constructible” arrangement of pseudolines with realization space equivalent to the open semi-algebraic set defined by the system. “Constructible” means “coming from a free construction using ruler”. It has an order on the pseudolines such that every next is incident to not more then two points in general position defined as intersections of pervious pseudolines. Then such an arrangement can be perturbed to simple with the equivalent realization space by an inductive perurbation trick applied to pseudolines in the reverse order of the construction. The trick was introduced independently by B. Sturmfels and N. White Constructing uniform oriented matroids without the isotopy property, see also Shor Lemma 4 and around.

The answer is indeed No. The most economic example up now i think is mentioned Suvorov's. Independently examples was constructed by P. Mani, B. Jaggi, B. Sturmfels, N. White "Uniform oriented matroids without the isotopy property"
My perhaps first by time example of simple line arrangement without isotopy property has 19 lines .

I will use language of line arrangements.

As @joseph-orourke correctly guessed in reply to @j.c. the stronger statement holds:

Any open basic semi-algebraic set (i.e the set defined by strict polynomial inequalities with integer coefficients) can be realized up to certain trivial stabilization by realization space of a simple pseudolines arrangement.

It was the initial target theorem due to some applications to Morse theory of vector functions. Reference is N. E. Mn"ev, The universality theorems on the classification problem of configuration varieties and convex polytopes varieties Theorem A, statement 2). This old paper only briefly describes the content of unpublished phd in Russian. Unfortunately this moment was not explicitly covered in later improvements and expositions of the proof. But it is really a simple addition to the main construction modernized by P.Shor, H. Gunzel and U. Richter-Gebert. The point is that having as an input a system of strict algebraic inequalities the general machine produces “constructible” arrangement of pseudolines with realization space equivalent to the open semi-algebraic set defined by the system. “Constructible” means “coming from a free construction using ruler”. It has an order on the pseudolines such that every next is incident to not more then two points in general position defined as intersections of pervious pseudolines. Then such an arrangement can be perturbed to simple with the equivalent realization space by an inductive perurbation trick applied to pseudolines in the reverse order of the construction. The trick was introduced independently by B. Sturmfels and N. White Constructing uniform oriented matroids without the isotopy property, see also Shor Lemma 4 and around.