The real-algebra tag has no wiki summary.

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### Sum of Squares Length of a Product

Let $n \geq 2$. Let $g_1, \ldots , g_{n-1} \in \mathbb{R}[x_1,\ldots,x_n]$ such that $q=g_1^2+\ldots +g_{n-1}^2$ is not divisible by $p=x_1^2+\ldots +x_n^2$. Let $m \geq 1$ be the smallest integer ...

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### Positivstellensatz for non-polynomial term

Can we use Positivstellensatz (P-satz) below for a non-polynomial term?
P-satz:
Let $R$ be real closed field. Let $f,g,h$ be finite families of polynomials in $R[X_{1} ,...,X_{n}]$. Denote by P the ...

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### Compactness of a semi algebraic set

Suppose I have a polynomial $p\in R[x_1,\ldots,x_n]$ and I look at the set $S:=\{ x\in R^n : p(x)\geq 0\}$. Are there algebraic certificates on $p$ that will certify that $S$ is compact?

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### Integer-valued power towers

$$\text{Let }f_n(a)=\underbrace{2^{2^{.^{.^{.^{2^a}}}}}}_{\text{$n$ 2s}}.$$
Obviously, $f_n(a)$ is an integer for every positive integer $n$ and non-negative integer $a$.
Are there any positive ...

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### Simple criterion to verify that the real zeros are an irreducible algebraic set

Given an irreducible polynomial $p$, its set of real zeros might be a reducible algebraic set. For example: $p=(x^2-1)^2 +(y^2-1)^2$.
Is there a simple sufficient condition on $p$ so that its real ...

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### Sums of Squares and Totally Positive Numbers

In Van der Waerden B L. Algebra Vol.I[M]. Springer, 2003, Pro. Waerden announced in page 256 that if an element $\gamma$ of a formally real field K is not a sum of squares, there exist an ordering of ...

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### Level of a commutative ring and its quotient field.

Reading Lam's Introduction to Real Algebra, he remarks that:
For a Dedekind domain $A$ with quotient field $F$, then $s(A)$ is either $s(F)$ or $s(F) + 1$. Furthermore, $s(A)$ is either $\infty$, ...

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### Smallest real closed field realizing all cuts of the rational numbers

Let $K$ be a real closed field of transcendence degree 1 over $\mathbb{R}$.
It is not difficult to see that $K$ has the following "minimality property": Whenever $L$ is a real closed field that ...

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### dense real closed fields

Let M_0\subseteq M_1 be two real ordered fields where M_0 is dense in M_1. Then is the real closure of M_0 dense in the real closure of M_1?

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### Real closed field+ model thoery

Is it true that every real closed field can be elementarily embedded in some other real closed filed with the same Archimedean classes (I mean in a proper extension)?
Can for example real numbers be ...

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### two real closed fields- algebraic elements

If R_1\subset R_2 are two real closed fields (R_2 is an extension of R_1), then is it always the case that R_1 contains {R_2}_alg; By the latter I mean algebraic elements of R_2.

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### an algebraically closed field definable in a real closed field

Is it true that an algebraically closed field $k$ definable (in the model theory sense) in a real closed field $\mathcal R$ is the algebraic closure $\mathcal{R}^{alg}=\mathcal{R}(\sqrt{-1})$?
...

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### normal domains with algebraically closed quotient field

I am looking for an integral domain $A$ with the following properties:
$A$ is not integrally closed
$A$ has a quotient field $K$ that is algebraically closed and that has characteristic 0
There is ...

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### Artin Schreier Theorem for Rings

This has been in my mind for quite some time. Looking at Artin Schreier Theorem for fields:
If L is a field and K its algebraic closure and if 1< [K:L] < infinity then L=K[i] and L is a real ...