Questions tagged [real-algebra]

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-2 votes
1 answer
65 views

Partial derivative in terms of Kronecker delta and the Laplacian operator [closed]

How can the following term: $$ T_{ij} = \partial_i \partial_j \phi$$ be written in terms of Kronecker delta and the Laplacian operator $\mathbin\bigtriangleup = \nabla^2$? I mean is there a relation: $...
4 votes
2 answers
305 views

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 $\infty$, $2^{...
5 votes
1 answer
242 views

Pythagorean numbers of real cyclotomic fields

Pythagorean number of a totally real field $\mathbb{K}$ is the minimal number $N$ of squares $t_k^2$ required to represent a totally positive $0\leq x\in \mathbb{K}$ as $x=\sum_{k=1}^N t_k^2$, where $...
5 votes
1 answer
685 views

Do real analytic functions on $\mathbb{C}\mathbb{P}^n$ form a Noetherian ring?

Question: Is the ring of real-analytic functions on $\mathbb{C}\mathbb{P}^n$ (real valued) a Noetherian ring? References or counterexamples are welcome. I know that the ring of germs of holomorphic ...
2 votes
0 answers
125 views

Topology of the set of polynomials with bounded real algebraic varieties (inside the v. s. of polynomials in $n$ variables and up to degree $d$)

Set $x=(x_{1}, \dots, x_{n}).$ Consider the set $\mathbb{R}[x]_{d}$ of polynomials with coef. in $\mathbb{R}$ in $n$ variables up to degree $d.$ This set can be seen as a finite-dimensional vector ...
2 votes
0 answers
117 views

A structure sheaf for real analytification of semialgebraic sets in the context of signed tropicalization

Let $X=Spec(A)$ be an affine scheme, where $A$ be a commutative algebra over a non-archimedean valued field $K$. Assume that $K$ is a real closed field with the unique ordering $<$, which should be ...
2 votes
0 answers
158 views

Representations in Archimedean quadratic modules

Let $\mathbb R [X] = \mathbb R [X_1,\dots,X_n]$ and $\Sigma[X] = \big\{ \, f \in \mathbb R[X] \mid \exists r \in \mathbb N, \ g_i \in \mathbb R[X] \colon f = g_1^2 + \dots + g_r^2 \,\big\}$ denote ...
7 votes
1 answer
728 views

Non-normal domain with algebraically closed fraction 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 an ...
1 vote
1 answer
64 views

Positivity in extensions of ordered fields

Let $F$ be an ordered field and $f\in F[X]$ be a polynomial such that $f(x)>0$ for all $x\in K$. Is it possible that there is an extension $L\supseteq K$ of ordered fields and $y\in L$ such that $f(...
0 votes
0 answers
86 views

Tensor product of preordered rings

All rings in this post are commutative, unital, and contain $\frac{1}{2}$. To study "real" properties of a ring $R$, one is often interested in the orderings which exist on fraction fields of ...
5 votes
1 answer
296 views

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?
0 votes
1 answer
243 views

Classification of finite-dimensional real super C*-algebras

The title says it all. I feel like one should be able to find this somewhere, but every time I try to google, I just get results for "super Lie algebras". Does anybody know a reference? I am not so ...
3 votes
0 answers
53 views

Points of intersection of summand of sums of squares of real polynomials

$\newcommand\R{\mathbb R} \newcommand\Q{\mathbb Q} $I am thinking of something related to Blekhermans 2012 paper Nonnegative Polynomials and Sums of Squares (Journal of the AMS, 25, 2012, 617-635). ...
4 votes
1 answer
356 views

Rotatable matrix, its eigenvalues and eigenvectors

We say that a real matrix is rotatable iff after turning it clockwise on $90^{\circ}$ it doesn't change. I'm interesting about eigenvalues and eigenvectors (belonging to non-zero eigenvalues) of such ...
7 votes
1 answer
274 views

Uniquely ordered commutative rings

I am wondering whether there are reasonable necessary and/or sufficient conditions to dedice whether a commutative ring can be uniquely ordered (like for instance $\mathbb{Z}$) or not. In the field ...
4 votes
2 answers
575 views

Proper embedding of a real closed field into another real closed field with the same Archimedean classes

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

Group of units of a valuation

Let K be a field. Then a subring R of K is called a valuation ring if for all $x \in K^*,$ either $x \in R$ or $x^{-1} \in R$ (or both). It can be shown that for any valuation $v$ on $K,$ the ring $\...
12 votes
2 answers
488 views

"Pythagoras number" for integral matrices

It is classically known that every positive integer is a sum of at most four squares of integers, i.e. every sum of squares of integers is a sum of four squares of integers. Now consider a symmetric $...
2 votes
1 answer
148 views

Polynomial degree comparison of Nullstellensatz and Positivstellensatz over real algebraic sets

Suppose we have a (finite) system of polynomials $P = \{ p_i \} \subseteq \mathbb{R}[x_1, \ldots, x_n]$. Then it is well known by the Nullstellensatz that either $P$ has a simultaneous zero over $\...
1 vote
0 answers
91 views

vector space of ternary forms with real rooted property

Let $V \subseteq \mathbb{R}[x,y]_d$ be a two dimensional linear subspace of the vector space of bivariate forms of degree $d$. For each degree $d$ we can find such subspaces with the property that ...
8 votes
1 answer
677 views

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 $\mathcal{R}^{alg}=\mathcal{R}(\sqrt{-1})$? UPDATE:...
3 votes
1 answer
156 views

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 ...
1 vote
0 answers
111 views

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 ...
5 votes
1 answer
211 views

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 ...
13 votes
0 answers
407 views

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 ...
0 votes
1 answer
228 views

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 ...
5 votes
1 answer
399 views

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 ...
0 votes
1 answer
233 views

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?
1 vote
2 answers
241 views

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.
4 votes
0 answers
692 views

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 ...