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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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 ...
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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 ...
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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 ...
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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 ...
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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\...
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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 ...
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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 ...
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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\...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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. ...
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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 ...
- 1,571
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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 ...
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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 ...
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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 $f^{(n)}(x)\geq 0, \ \forall x\in \mathbb R$. ...
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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 ...
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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 ...
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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 ...
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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 \...
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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,\...
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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 ...
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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 ...
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Is there a general method for finding real solutions to this class of systems of equations?
Fix a positive integer $n$. Let $\mathbf f=(f_1,\ldots,f_n)$ such that for each positive integer $i\le n$:
the function $f_i:\Bbb R^{n-1}\to\Bbb R$ is a polynomial function with real [real algebraic?]...
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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 ...
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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 ...
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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.
...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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On the properties of sum-of-squares polynomials
Definition 1. If a multivariate polynomial $f$ can be written as a finite sum of squared polynomials, i.e., $f(x)=\sum_{i = 1}^n g_i^2(x)$, then $f$ is SOS.
Definition 2. If an $n$-variate polynomial ...
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Totally positive/negative units in preordered rings with bounded inversion
Let $A$ be a preordered ring (or $\mathbb{R}$-algebra or $\mathbb{Q}$-algebra).
Say that ${a \in A}$ is totally positive if for every morphism ${f : A \rightarrow \mathbb{R}}$ of ordered algebras, ${f ...
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Defintion of a real algebraic space in Atiyah's K-theory and reality
In Atiyah's paper "K-theory and reality", p. 370, there is an example of a "real" algebraic space. Given the complex projective space $X=P(C^n)$, one considers the standard line-...
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The derivative of a semialgebraic map is semialgebraic
Coste's notes on semialgebraic geometry have the question:
If $f:U \to \mathbb{R}$ is semialgebraic, with $U$ an open semialgebraic set, then each partial derivative $\dfrac{\partial f}{\partial x_{i}...
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Extension theorem over reals
Is there an equivalent of the following theorem from Cox, Little & O'Shea over reals?
Definition.
Given $I=\left\langle f_{1}, \ldots, f_{s}\right\rangle \subseteq k\left[x_{1}, \ldots, x_{n}\...
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Are these definitions of dimension for a real algebraic variety all equivalent?
I would like to believe that the following notions of dimension are all equivalent for an a real algebraic variety in $\mathbb{R}^n$.
Lower Minkowski dimension
Upper Minkowski dimension
Hausdorff ...
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Dimension of a semialgebraic set equals the dimension of its closure.
I am trying to prove that if $X \subset \mathbb{R}^{n}$ is semialgebraic, then $\operatorname{dim}X = \operatorname{dim}\overline{X}$, where the dimension of a semialgebraic set is the supremum of the ...
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Product of semialgebraic sets is semialgebraic
I am trying to prove that the product of semialgebraic sets is semialgebraic.
If $X \in \mathbb{R}^{n}$ and $Y \subset \mathbb{R}^{m}$ are semialgebraic, I can't see the polynomial conditions involved ...
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Real radical ideal which is not prime
Terminology: Let $I\subset\mathbb{R}[X_1,...,X_n]=:A$ be an ideal. We call $\sqrt[\mathbb{R}]{I}:=\{f\in A\ |\ \exists k\in\mathbb{N},\ g_1,...,g_m\in A:\ f^{2k}+\sum_{i=1}^mg_i^2\in I\}$ the real ...
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Semialgebraic set technical lemma
Suppose that $U\subseteq \mathbb{R}^3$ is real-semialgebraic set such that $U'\times \mathbb{R}\subseteq \overline{U}$ where $U'$ is an infinite set. I want to show that there exist $x,y$ such that ...
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Parametrizing an algebraic curve
Suppose $C\subset \mathbb{R}^3$ is a path-component of an irreducible real-algebraic curve. Is there a smooth parametrization that covers all but possibly one or two points of $C$? If not, is there a ...
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Number of intersection points of plane and algebraic curve
Suppose $C\subseteq \mathbb{R}^3$ is an irreducible real-algebraic curve of degree $k$ and $P\subset \mathbb{R}^3$ is a plane. Suppose that $C$ intersects $P$ finitely many times. What is the best way ...
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Algebraic curve contained in plane or only intersects it finitely many times
Suppose that $C\subset \mathbb{R}^3$ is an irreducible real-algebraic curve that meets a plane infinitely many times. Is it true that $C$ must be contained in this plane?
I'm working on a paper that's ...
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Semi-algebraic set has nonempty interior relative to Zariski closure
Suppose $X\subseteq \mathbb{R}^3$ is a real semi-algebraic set. Consider the subspace topology induced by the Euclidean topology on the Zariski closure $\overline{X}.$ Can we guarantee that $X$ has ...
- 4,534
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Structure of Closed Semialgebraic set
I am trying to prove the following, from Benedetti and Risler's book:
The "above proposition" is:
It seems an easy proposition that boils down to taking the complement of the complement of ...
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