Is every subset of ${\mathbb R}^n$ that is both open and semi-algebraic equivalent to one defined only using strict inequalities of polynomials?

This looks like an undergraduate question, but it comes from the research context below and I have so far got no response.

I think it's true, but my intuition for algebraic geometry was never very good, even at school level.

If this is true then so is my main question below, because

• every predicate in my bounded language is semi-algebraic, since
  $\forall x:[a,b].\phi x\iff\forall x:{\mathbb R}.x<a\lor\phi x\lor b<x$ and
  $\exists x:[a,b].\phi x\iff\exists x:{\mathbb R}.a\leq x\land\phi x\land x\leq b$;

• predicates in my bounded language define open subsets and the product projection is an open map;

• the Tarski-Seidenberg theorem eliminates quantifiers;

• the result may involve equations or non-strict inequalities, but since the projected subset is open, the conjecture eliminates these.

The book Algorithms in real algebraic geometry by Saugata Basu, Richard Pollack and Marie-Françoise Roy (Springer 2006) is freely and legally available online. It looks very well written and relevant to this question, but Remark 3.2 comes tantalisingly close without answering it.

If the answer to my question is positive then it would be good to involve semi-algebraic geometers in the community that I describe below.

If there is a counterexample, it would provide a simple way of explaining to referees etc that that subject is not relevant to mine.

Any bounded predicate has a partner obtained as a "de Morgan dual", ie by switching $$ +/-,\quad >/<,\quad \top/\bot,\quad \min/\max,\quad \sup/\inf,\quad \land/\lor,\quad \forall/\exists. $$

Can this theorem be adapted to my language with strict inequalities and bounded quantifiers? It is plausible, because my bounded predicates certainly do define open subsets.

If the polynomial is of degree $\leq6$ in the quantified variable then there are formulae (in the other variables) for the zeroes of its second derivative. These (and the endpoints) provide the extreme values, so substituting them eliminates the quantifier in favour of finite disjunctions or conjunctions. This is at the cost of using square and cube roots, but I believe that there are old ways of eliminating these too.

Is there a way of doing this without the formulae for the roots or for higher-degree polynomials?

This theorem says that quantifiers over real variables can be eliminated from classical first order formulae built from equations and inequalities between polynomials with rational coefficients, ie in a language with ${=},{\leq},{\geq},{\neq},{<},{>},{\land},{\lor},{\lnot},{\Rightarrow}$ and $\forall x:{\mathbb R}$ and $\exists x:{\mathbb R}$.

Is there a similar result for just ${\neq},{<},{>},{\land},{\lor}$ and $\forall x:[a,b]$ and $\exists x:[a,b]$, ideally a constructive proof?

I am asking this towards the research goal of having a feasible way of computing Cauchy sequences from Dedekind cuts.

The research question is a conspicuous gap in the research community that I describe below, so since the answer to the textbook question above seems to be yes, I would like to invite semi-algebraic geometers to join that community.

The book Algorithms in real algebraic geometry by Saugata Basu, Richard Pollack and Marie-Françoise Roy (Springer 2006) is freely and legally available online. It looks very well written and relevant to this question, albeit classically, but Remark 3.2 comes tantalisingly close without answering it.

Any bounded predicate has a partner obtained as a "de Morgan dual", ie by switching $$ >/<,\quad \top/\bot,\quad \min/\max,\quad \sup/\inf,\quad \land/\lor,\quad \forall/\exists. $$

Can this theorem be adapted to my language with strict inequalities and bounded quantifiers? It is plausible, because my bounded predicates certainly do define open subsets.

If this is true then

• every predicate in my bounded language is semi-algebraic, since
  $\forall x:[a,b].\phi x\iff\forall x:{\mathbb R}.x<a\lor\phi x\lor b<x$ and
  $\exists x:[a,b].\phi x\iff\forall x:{\mathbb R}.a\leq x\land\phi x\land x\leq b$;

• predicates in my bounded language define open subsets and the product projection is an open map;

• the Tarski-Seidenberg theorem eliminates quantifiers;

• the result may involve equations or non-strict inequalities, but since the projected subset is open, the conjecture eliminates these.

I think it's true, but my intuition for algebraic geometry was never very good, even at school level.

If this is true then so is my main question below, because

• every predicate in my bounded language is semi-algebraic, since
  $\forall x:[a,b].\phi x\iff\forall x:{\mathbb R}.x<a\lor\phi x\lor b<x$ and
  $\exists x:[a,b].\phi x\iff\exists x:{\mathbb R}.a\leq x\land\phi x\land x\leq b$;

• predicates in my bounded language define open subsets and the product projection is an open map;

• the Tarski-Seidenberg theorem eliminates quantifiers;

• the result may involve equations or non-strict inequalities, but since the projected subset is open, the conjecture eliminates these.

I am interested in the following language of predicates on ${\mathbb R}^n$, which I am calling bounded:

Is every subset of ${\mathbb R}^n$ that is both open and semi-algebraic equivalent to one defined only using strict inequalities of polynomials?

This looks like an undergraduate question, but it comes from the research context below and I have so far got no response.

If this is true then

• every predicate in my bounded language is semi-algebraic, since
  $\forall x:[a,b].\phi x\iff\forall x:{\mathbb R}.x<a\lor\phi x\lor b<x$ and
  $\exists x:[a,b].\phi x\iff\forall x:{\mathbb R}.a\leq x\land\phi x\land x\leq b$;

• predicates in my bounded language define open subsets and the product projection is an open map;

• the Tarski-Seidenberg theorem eliminates quantifiers;

• the result may involve equations or non-strict inequalities, but since the projected subset is open, the conjecture eliminates these.

The book Algorithms in real algebraic geometry by Saugata Basu, Richard Pollack and Marie-Françoise Roy (Springer 2006) is freely and legally available online. It looks very well written and relevant to this question, but Remark 3.2 comes tantalisingly close without answering it.

If the answer to my question is positive then it would be good to involve semi-algebraic geometers in the community that I describe below.

If there is a counterexample, it would provide a simple way of explaining to referees etc that that subject is not relevant to mine.

My language

I am interested in the following language of predicates on ${\mathbb R}^n$, which I am calling bounded:

Context

Dedekind cuts to Cauchy sequences

The Tarski-Seidenberg theorem

Research Communities