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Questions tagged [real-algebraic-geometry]

Real algebraic geometry is the study of algebraic geometry over the real numbers, or more generally formally real (esp. real closed) fields. Problems in this tag may require a mix of methods from algebraic geometry and techniques from o-minimal (esp. semialgebraic) geometry.

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Checking if a polynomial ideal is real

I'm working with a polynomial ideal $I \subset \mathbb{R}[s_1,c_1, \ldots, s_n, c_n]$ generated as $I = \langle s_1^2+c_1^2-1, \ldots, s_n^2+c_n^2-1\rangle$ and looking to show that this ideal is real....
pwensing's user avatar
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Polynomial formula for orthogonal vector in odd dimensions [duplicate]

I have been thinking about this problem recently. In 2 dimensions there is an easy formula for a nonzero vector orthogonal to a given vector $(x, y)$, namely $(-y, x)$. By taking pairs of coordinates, ...
EthanK's user avatar
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2 votes
1 answer
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Partial derivatives "split" over $\mathbb R$

Kaplansky's book Fields and Rings, page 30, implicitly contains the following question. For which fields $K$ does the following statement hold true for every polynomial $g\in K[X]$: If $g$ splits ...
Daniel W.'s user avatar
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2 votes
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A semi-algebraic set in $\mathbb{R}^d$ has dimension $d$ if and only if its interior is non empty

I would like to prove the following result : A semi-algebraic set in $\mathbb{R}^d$ has dimension $d$ if and only if its interior is non empty The dimension here has to be understood in the semi ...
G2MWF's user avatar
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6 votes
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113 views

Can we determine into how many regions the zero set of a polynomial divides the plane?

Disclaimer: If you look at my other questions, you will see that I am still learning the basics of algebra. I am well aware that the answer to this question is likely to go beyond my current level of ...
paulina's user avatar
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Example of semi algebraic subset over a real closed field

I am trying to find interesting example of semi algebraic subset over a real closed field other than those we have by considering $\mathbb{R}^n$. I tried to construct an example involving matrices. ...
G2MWF's user avatar
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Semi algebraic sets and smooth manifolds

I would like to prove a result that I cannot see why it could be wrong in order to fix the idea. Here is the result : Let $A\subset\mathbb{R}^{n}$ be a semi algebraic set which is a smooth ...
G2MWF's user avatar
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1 vote
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Link between semi-algebraic dimension and vector space dimension

I am new to semi-algebraic geometry and I have encountered a very pleasant proposition to define the semi-algebraic dimension of a semi-algebraic subset $A$ of $\mathbb{R}^n$. This definition is based ...
G2MWF's user avatar
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0 votes
1 answer
29 views

Difficulty to prove that a set is semi algebraic

I consider for $i\in\{1,…,N\}$ the subset $X_i\subset\mathbb{R}^{n}$ that are convex and compact and semi algebraic. We denote by $X=\Pi_{i}^{N}X_i$ the Cartesian product of the $X_i$ which is still ...
G2MWF's user avatar
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rational map by F. Mangolte

I'm reading Real Algebraic Varieties by F. Mangolte. Definition 1.3.22 (in the book) If $X$ and $Y$ are algebraic varieties over a base field $K$ a rational map $\phi:X\dashrightarrow Y$ is an ...
isz's user avatar
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misunderstanding on real algebraic varieties

Bochnak-Coste-Roy's book "Real Algebraic Geometry" (1998) is probably the main reference on this subject. I am probably misunderstanding something very fundamental as I can apparently find ...
Hair80's user avatar
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2 votes
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64 views

Notation $\text{RSpec}(K)$ for space of orderings of a field $K$

I'm reading notes from commutative algebra by Pete Clark. In one of the examples of Galois connections he introduces notation $\text{RSpec}(K)$ for the set of all total orders on a field $K$, and ...
Jakobian's user avatar
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1 vote
1 answer
75 views

Semi-numeric solutions to a system of polynomial equations when a Groebner basis is too complicated

I have a system of polynomial equations with rational coefficients and I would like to find real solutions, if they exist. The system has $n\sim 10$ unknowns, $n$ equations with degree $\sim 2n$ and ...
Christian Chapman's user avatar
1 vote
0 answers
41 views

Homotopy equivalence vs deformation retract for analytic spaces

I know that there are many examples of spaces $Y\subset X$ such that $X$ and $Y$ are homotopy equivalent but there is no deformation retract of $X$ to $Y$ (e.g., Does homotopy equivalence to a ...
MathBug's user avatar
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Connected components of the zero set of multilinear polynomials

Let $p\in \mathbb{R}[x_1, \ldots, x_n]$ be a real (non-constant) multilinear polynomial of degree $\deg p=d$, then its real zero-set $V_{\mathbb{R}}(p)$ is non-empty and has dimension $n-1$ by the ...
Simon's user avatar
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1 answer
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Proving a system of quadratic forms has no (non-zero) solutions

A system of homogeneous linear equations always has the solution $ x=(0,\dots, 0) $. Suppose we have a system of $ n $ homogeneous linear equations in $ k $ variables. If $ k > n $ then there will ...
Ian Gershon Teixeira's user avatar
1 vote
1 answer
51 views

System of $ n $ simultaneously diagonal real quadrics in $ n+1 $ variables has all solutions real

A single isotropic real quadric in 2 variables always has 2 (projective) real solutions. See Zero set of system of two real quadratic forms for the explicit form. I've noticed that a system of 2 ...
Ian Gershon Teixeira's user avatar
1 vote
0 answers
58 views

When are the zeroes from Bezout's theorem real/rational/integral

Suppose I have $n$ quadratic homogeneous polynomials $f_1, \cdots, f_n$ in $n+1$ variables over the field $\mathbb{C}$. Bezout's theorem says that generically there will be $2^n$ common zeroes. I was ...
Eric Kubischta's user avatar
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Help me complete a proof about the enumeration of similar intersections of an algebraic surface with a hyperplane

This is something of a follow-up to this question that i posted almost a year ago: What are the conditions on two multivariate real polynomials $f$ and $g$ so that their zero sets are similar? It was ...
paulina's user avatar
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0 answers
20 views

Question on matrix spaces that is derived from several vector spaces

Suppose I have $m$ column vectors $x_i \in X \subset \mathbb{R}^n$, $i \in \{1,\ldots,n\}$. If I construct a matrix $\mathbf{x}= [x_1^T ; \ldots ; x_m^T]$, then what set does $\mathbf{x}$ live in ...
Acad's user avatar
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What is the minimal number of polynomial squares needed to represent any sum of any number of polynomial squares?

Let $Q^{(n, d)}$ be a set of real polynomials in $n$ variables of degree $d$, and $S_k^{(n, d)}$ the set of the sum of squares of any $k$ such polynomials: \begin{equation} S_k^{(n, d)} := \left\...
mrlovre's user avatar
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6 votes
1 answer
101 views

Proof of uniqueness of real-closure of an ordered field

I'm reading the proof of uniqueness of real-closure of an ordered field $F$, that is an algebraic extension $R$ of $F$ such that $R$ is real-closed and the unique order on $R$ extends that of $F$. I ...
Jakobian's user avatar
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0 votes
0 answers
29 views

Theorems of calculus for real-closed fields [duplicate]

Theorems such as Intermediate value theorem Rolle's theorem Mean value theorem Positive/negative derivative implies strictly increasing/decreasing Hold for polynomials over real-closed fields. If ...
Jakobian's user avatar
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1 vote
0 answers
60 views

Image of a 'narrow' set under a polynomial mapping is a proper semialgebraic subset.

Consider a set $X\epsilon=\{y^2 - \epsilon^2 x^2 \leq 0\} \subset \mathbb{R}^2, 0<\epsilon <<1$ , i.e. a narrow cone passing through the origin. I would like to prove some properties of $f(X\...
John Doe's user avatar
0 votes
1 answer
54 views

Real geometric variety and the real torus

Let $T^2$ be a real torus as an $\mathbb{R}$-affine variety. For example $$ T^2 = \{ (x_1,x_2,x_3,x_4) \in \mathbb{R}^4 | x_1^2 + x_2^2 -1 =0, x_3^2+x_4^2-1=0 \} .$$ Let $V$ be a real irreducible ...
Giacomo Bascapè's user avatar
6 votes
1 answer
131 views

Example of minimal parabolic k-subgroups that are not Borel subgroups

A subgroup $B < G$ of a connected algebraic group $G$ that is maximal among the solvable connected subgroups is called a Borel subgroup. A closed subgroup $P < G$ is parabolic if it contains a ...
Strichcoder's user avatar
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5 votes
1 answer
84 views

Algebraic varieties fulfilled by solutions of polynomial ODEs

Let's assume we have a two dimensional polynomial vector field of degree $d$ $$F: \mathbb{R}^{2}\rightarrow\mathbb{R}^{2}, \quad (x,y)\mapsto \begin{pmatrix}P(x,y), \\ Q(x,y)\end{pmatrix}$$ and we are ...
NicAG's user avatar
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1 vote
0 answers
77 views

Polynomials that are non-negative outside the $L_2$ unit ball

Let $\mathcal{P}$ be the set of all bivariate polynomials over the reals of degree at most two, that are non-negative outside the $L_2$ unit ball. Are all the polynomials in $\mathcal{P}$ one of the ...
Mathews Boban's user avatar
0 votes
5 answers
152 views

Finding integer roots of an integer quadratic form

Let $ Q\in \mathbb{Z}[x_1, \dots, x_n ] $ be an integer quadratic form. That is, $ Q $ is a homogeneous degree $ 2 $ polynomial with integer coefficients. Is there a good way to determine if $ Q $ has ...
Ian Gershon Teixeira's user avatar
1 vote
0 answers
87 views

Groebner basis for system of integral quadratic forms

I have a system of quadratic forms (homogeneous polynomials of degree $ 2 $) with integer coefficients. Each quadratic form is the trace of a product of matrices. I'm solving for the matrix entries. ...
Ian Gershon Teixeira's user avatar
1 vote
1 answer
38 views

connected components of birational real surfaces

Suppose $X$ and $Y$ are smooth real algebraic surfaces in $\mathbb P^3(\mathbb R)$. If X and Y are birational over the reals, then is it true that they also have the same number of connected ...
quantum's user avatar
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7 votes
1 answer
196 views

Real number known not to be a period

I am working a bit with problems in non-archimedean settings inspired by the famous periods conjecture by Kontsevich-Zagier. I was preparing a talk and wanted to give of background about the initial ...
Florian Felix's user avatar
0 votes
5 answers
197 views

Finding redundant equations in a underdetermined system of multivariate polynomial equations over $\Bbb R$

Starting from a geometric problem, I came up with a system of multivariate (many lines and points) polynomial equations where some equations are redundant (because they correspond to redundant ...
mmj's user avatar
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2 votes
1 answer
80 views

A function with weakly positive $n$-th derivative has at most $n$ roots – an inequality version of the Fundamental theorem of Algebra.

This is an attempt to generalize the result in [1]. Claim: Let $n\in \mathbb N$, and let $f:\mathbb R \to \mathbb R$ be such that its $n$-th derivative $f^{(n)}(x)\geq 0, \ \forall x\in \mathbb R$. ...
Pavel Kocourek's user avatar
8 votes
1 answer
305 views

A function with positive $n$-th derivative has at most $n$ roots – an inequality version of the Fundamental theorem of Algebra.

Claim: Let $n\in \mathbb N$, and let $f:\mathbb R \to \mathbb R$ be such that its $n$-th derivative $f^{(n)}(x)>0, \ \forall x\in \mathbb R$, then $f$ has at most $n$ roots. Context: The ...
Pavel Kocourek's user avatar
3 votes
0 answers
94 views

The polynomial $1 + x_1^2 + \dots + x_n^2$ cannot be written as a sum of $n$ rational squares of polynomials.

Artin (1927) settled Hilbert's 17th problem– any nonnegative polynomial can be written as a sum of squares of rational polynomials. Cassels (1964) proved that a if a polynomial admits an SOS ...
deej's user avatar
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2 votes
1 answer
56 views

Reference request for texts on a generalization of real algebraic geometry.

I know that the field of real algebraic geometry studies zero sets of real polynomials. I am looking for texts on a generalization of real algebraic geometry that talk about zero sets of classes of ...
user107952's user avatar
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0 votes
1 answer
326 views

Question regarding definition of semialgebraic set

I read from notes that a semi-algebraic set in $\mathbb{R}^n$ is a set that can be given by finitely many polynomial equalities and inequalities. It is a finite union of sets of the form $\{x \in \...
Chang Henry's user avatar
2 votes
2 answers
85 views

Is volume of semialgebraic sets definable over the language of ordered fields?

Let $ \mathcal{L}=\{0,1,+,\cdot,\le\} $ be the language of ordered fields and consider the theory of $\mathbb{R} $ in this language (i.e., the theory of real closed fields). Suppose $ \varphi(x_1,\...
sss89's user avatar
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1 vote
0 answers
67 views

Continuity of real polynomial roots within a component (non-zero dimension variety)

I have an applied problem that deals with real multivariate polynomials systems (real coefficients with real roots). If I understand this problem correctly, the (real) root numbers will remain ...
Kryvtsov's user avatar
  • 131
0 votes
1 answer
88 views

The Irreducibility of the Algebraic Set of a Sphere Over $\mathbb{R}$

I've been trying to show that the sphere $x^2+y^2+z^2=1$ is irreducible as an algebraic set over $\mathbb{R}$. By this I mean that it cannot be written as the union of two proper algebraic subsets. It ...
Dave's user avatar
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2 votes
1 answer
85 views

What are the conditions on two multivariate real polynomials $f$ and $g$ so that their zero sets are similar?

Basically what the title says. Let $f$ and $g$ be two polynomials with real coefficients of degree $m$, in $n$ real variables. The solutions of the equation $f(x_1,\ldots,x_n) = 0$ then define a ...
paulina's user avatar
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1 vote
0 answers
91 views

From high school level analytic geometry to elementary real algebraic geometry (reference request)

I want to learn some real algebraic geometry. My background is high school mathematics. Is there a textbook that provides a path from high school level analytic geometry to elementary real algebraic ...
jainemarie's user avatar
0 votes
0 answers
43 views

Arbitrary sum of squares of rationals

I am studying real closd fields for quantifier elimination of RCF and proved the following theorem. Let $F$ be a field. Then the following properties are equivalent: The field $F$ is real closed. ...
Gowexx's user avatar
  • 372
1 vote
0 answers
80 views

Number of equations and number of variables in system of polynomial equations

It is well known that if we have a system of linear equations of $n$ variables, we need $n$ equations to solve the system. I wonder if we can say the same thing for a system of polynomial equations of ...
hurrikale's user avatar
1 vote
0 answers
90 views

How can we know the Topology of $\{(x,y) \in \Bbb R ^2 : f(x,y)=0 \}$

Let $f \in \Bbb R[x,y]$ be a given polynomial and set $$V=\{(x,y)\in \Bbb R^2 : f(x,y)=0\}$$ How can we tell if $V\neq \emptyset$ ? How can we know if $V$ is compact? How can we tell if $V$ is ...
Marcos Martínez Wagner's user avatar
2 votes
1 answer
138 views

Polynomial Maps of Real Projective Varieties

As a preface I don't much background in this area and I think I am dealing with older (or currently non-standard) definitions. Definition: A polynomial $f \in \mathbb{R}[x_1, \ldots, x_n]$ is ...
Rishi Sonthalia's user avatar
2 votes
2 answers
171 views

Path connectedness equivalent to connected in euclidean topology for quasiprojective vareities?

Let $V$ be a quasiprojective real variety. My intuition tells me that this type of space with the Euclidean topology has the property that path-connectedness and connectedness are equivalent. Is this ...
quantum's user avatar
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3 votes
0 answers
70 views

Is there a useful classification of the homogeneous spaces for real Lie groups?

Let $G$ be a compact semisimple real Lie group. For the complex case there is a very deep theory connecting $G$-homogeneous spaces with irreducible representations of $G$. My question is: Is there an ...
smitke6's user avatar
  • 699
3 votes
0 answers
165 views

Triangulation Theorem for semialgebraic maps

Benedetti and Risler's "Real algebraic and Semi-algebraic sets" book on Semialgebraic Geometry has the following theorem: Theorem 2.6.14 Let $f:V \to Y$ be a continuous semialgebraic mapping ...
André's user avatar
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