Questions tagged [real-algebra]
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30 questions
-2
votes
1
answer
102
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:
$...
- 117
5
votes
1
answer
277
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 $...
- 14k
2
votes
0
answers
155
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 ...
- 573
2
votes
0
answers
125
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 ...
- 21
2
votes
0
answers
172
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 ...
- 31
1
vote
1
answer
65
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(...
- 193
0
votes
0
answers
94
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 ...
- 277
0
votes
1
answer
273
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 ...
- 3,001
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).
...
- 2,275
4
votes
1
answer
380
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
282
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 ...
- 131
5
votes
1
answer
777
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 ...
- 589
2
votes
0
answers
120
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 $\...
- 131
12
votes
2
answers
499
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 $...
- 3,031
2
votes
1
answer
152
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 $\...
- 539
1
vote
0
answers
92
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 ...
- 3,031
3
votes
1
answer
159
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 ...
- 3,031
1
vote
0
answers
113
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 ...
- 11
5
votes
1
answer
330
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?
- 65
13
votes
0
answers
413
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 ...
- 239
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 ...
- 61
0
votes
1
answer
241
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 ...
- 3
4
votes
2
answers
308
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^{...
- 143
5
votes
1
answer
418
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 ...
- 328
0
votes
1
answer
236
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?
- 365
4
votes
2
answers
594
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 ...
- 365
1
vote
2
answers
246
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.
- 365
7
votes
1
answer
696
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:...
- 4,070
7
votes
1
answer
746
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 ...
- 2,275
7
votes
0
answers
769
views
Artin-Schreier Theorem for Rings
This has been in my mind for quite some time. Looking at the Artin-Schreier Theorem for fields:
If $L$ is a field and $K$ its algebraic closure and if $1< [K:L] < \infty$ then $K=L[i]$ and $L$ ...
- 2,275