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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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 ...
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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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Semialgebraic Morse-Sard Theorem - why are the critical points of a semi-algebraic map again semi-algebraic?

2.5.12 Exercise (Semi-algebraic Morse-Sard theorem) Let $f:M\to N$ be a $C^\infty$ semi-algebraic map between semi-algebraic submanifolds of $\Bbb R^n$ and $\Bbb R^m$, respectively. Set $$C = \{x\in M;...
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Closed semialgebraic subset of $\mathbb{R}^2$

I'm trying to solve the following problem from exercise 2.13 of Michel Coste's An introduction to Semialgebraic Geometry (October 2002) [PDF]. Let $S$ be a closed semialgebraic subset of the plane ...
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Ordering of R(X) in real algebraic geometry

I am reading Bochnak, Coste, Roy's real algebraic geometry and the definition of ordering on $R(X)$ is not clear to me why/how can the authors write the polynomial $P(X)=a_nX^n+a_{n-1}X^{n-1}+\cdots+...
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Convex Hull of a Variety in Real Space

I am a physicist currently working on a question posed as part of an algebraic geometric description of a physical set: I did not find a question that is closely related to what I am searching for yet,...
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Smoothness of real analytic space

Consider $\mathbb{R}^2$ with the polynomial $P(x,w):=w^2+xw+x$. The zero set $N\left(P\right)$ of $P$ is a real analytic submanifold of $\mathbb{R}^2$. That is, the stalks of $\mathcal{A}_D/J_{N(P)}$ ...
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Book/notes on Real Analytic Functions.

I am currently working with Real Algebraic Geometry, and I'm needing references for Real Analytic Functions but I'm having a hard time finding them. Does anyone know a good book/set of notes on the ...
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Why is the boundary of spectrahedra “more pointy” at matrices of lower rank?

In the following expository article about spectrahedra, it is established informally that the boundary of spectrahedra is “more pointy” at matrices of lower rank. Cynthia Vinzant, What is a... ...
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On irreducible algebraic sets

I have the following maybe trivial quesion regarding real algbraic sets: Suppose $V\subset \mathbb{R}^n$ is an irreducible real algebraic variety (i.e. it can not be written as union of two algebric ...
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Algorithm to determine whether a system of multivariate polynomials has a real solution

Say I have a system of multivariate polynomials, $[P_{1}, ... ,P_{n}]$. I'm interested in determining whether it has a real-valued solution, but don't necessarily want to find any solutions, or ...
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Is the zero set of a real polynomial union of smooth manifolds? [duplicate]

Question: If I have a non-zero polynomial $P(x_1,\ldots,x_n)$ defined on $\mathbb R^n$, is it true that the zero set $$Z(P) = \{x \in \mathbb R^n: P(x_1,\ldots,x_n) = 0\}$$ can be written as finite ...
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Which Ideals and varieties correspond over $\mathbb{R}$?

We know that $\mathcal{I(V(}I)) = \sqrt{I}$ when the underlying field is algebraically closed. So, algebraic sets and radical idels correspond. The question that I wonder about is how much relation of ...
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How to create an Algebraic Variety/Polynomial Subject to Topological Constraints

I've had this question for a while and haven't seen anyone else answer it to my knowledge. Here goes: Construct all degree $d$ algebraic varieties in $n$ independent variables: $x\in \mathbb{R}^n$, $[$...
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2 votes
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Submanifolds and real algebraic sets

Nash-Tognoli theorem states that any real compact smooth manifold $A$ is diffeomorphic to a smooth real algebraic set. If $B$ is a compact submanifold of $A$, does this diffeomorphism $f$ induce a ...
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Number of connected components of zero set of polynomial with bounded number of terms

Suppose $f:\mathbb{R}^d\to\mathbb{R}$ is a polynomial of degree $\ell$. Then, the number of connected components of its zero set $\{a\in\mathbb{R}^d : f(a) = 0\}$ is bounded by roughly $\ell^{d}$. I'...
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Can every semi-algebraic set be written as a finite union of disjoint Nash submanifolds?

Can we write every semi-algebraic set $A \subset \mathbb{R}^m$ as the disjoint union of finitely many Nash submanifolds of $\mathbb{R}^m$? By Nash submanifolds are meant those manifolds that are ...
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An almost complex structure on the real 2-sphere $S^2$

If $R:=\mathbb{R}[x,y,z]/(x^2+y^2+z^2-1)$ and $S^2:=Spec(R)$ is the real 2-spere, a classical result of Borel and Serre says that the only real spheres with an almost complex structure is $S^2$ and $S^...
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Points at distance one from an algebraic subset of $\mathbb R^n$

Let $n$ be an integer $\ge2$; let $X_1,\ldots,X_n$ be indeterminates; let $S$ be a subset of the polynomial ring $\mathbb R[X_1,\ldots,X_n]$; let $V(S)$ be the set formed by the points of $\mathbb R^n$...
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Using Gröbner bases and Cylindrical Algebraic Decomposition to solve real polynomial systems

I'm working on a project that involves solving systems of multivariate polynomial equations over the reals (and find their real solutions). Assuming that a system has a finite number of complex ...
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How to determine the number of critical points a polynomial scalar field has?

Consider the function $$f(x,y) = x^3 + 3y - y^3 - 3x$$ How would I be able to determine the number of critical points $f(x,y)$ has? I know critical points will exist if $\nabla f(x,y) = 0$ or $f_x(x,y)...
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Can any polynomial in $\Bbb R [x]$ be written as the difference of two sum-of-squares polynomials in $\Bbb R [x]$?

Let $f \in \Bbb R[x_1,\dots,x_n]$ be an arbitrary polynomial, not necessarily non-negative. Are there always two sum-of-squares (SOS) polynomials $g, h \in \Bbb R[x_1,\dots,x_n]$ such that $f=g-h$? If ...
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isolated point of a real plane curve

$\newcommand\C{\mathcal C}$Let $\C:f(x,y)=0$ be a real plane algebraic curve, we will also assume that $\C$ has real points and in particular that $P=(0,0)$ is a point of $\C$. Let $f_d(x,y)$ be the ...
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Relationship between Number of Connected Components and number of real solutions?

Let $V$ be a a real variety in $\mathbb{R}^n$, defined by the solution set of a real polynomial system $f_i(x_1,x_2,...,x_n)=0, i=1,2,...,m$. #1. We can say the number of connected components of $V$ ...
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Can a real plane curve have a singular complex point but no singular real points?

Let $p(x,y)\in\mathbb{R}[x,y]$. Is it possible that $\partial_xp(x,y)=\partial_yp(x,y)=p(x,y)=0$ has a solution $(x,y)\in\mathbb{C}^2$ but no solutions $(x,y)\in\mathbb{R}^2$? In other word, can you ...
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1 vote
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What can replace "dimension $0$" to measure discreteness of varieties over $ \mathbb{R} $?

Consider the algebraic set $Z(x^2 + y^2) \subseteq \mathbb{R}[x, y]$. Set $A = \mathbb{R}[x, y]/(x^2 + y^2)$. In this ring, we have a chain of prime ideals $(0) \subseteq (x,y) $: the ring has ...
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Regular rational functions over real projective lines and projective spaces.

In Algebraic Geometry one usually says that "working with algebraically closed fields makes life easier". Today I stumbled over one instance of this: suppose $X$ is an irreducible projective ...
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Do exponents in tropical polynomials have to be integers?

Recently I've been learning about tropical geometry, and every time I see a definition of a tropical polynomial in e.g. $k$ variables $x_1,x_2,...,x_k$ such as $\bigoplus_{i=1}^n a_i x_1^{b_{i1}}x_2^{...
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When is a multivariate cubic polynomial mapping $\Bbb{R}^n$ to $\Bbb{R}^n$ both one-to-one and onto?

In one dimension, a cubic polynomial mapping $\Bbb{R}$ to $\Bbb{R}$ $$y = A + Bx + Cx^2 + Dx^3$$ is one-to-one and onto when its derivative $y'(x) = B + 2Cx + 3Dx^2$ has less than two zeros, i.e., $4C^...
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2 votes
1 answer
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How to solve $x^{T}Ax = 0$?

Given matrix $A \in \mathbb{R}^{n \times n}$, how do I solve $x^{T}Ax = 0$ for $x \in \mathbb{R}^n$? Obviously, a zero vector is always a solution and if $A$ is positive or negative definite there is ...
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Minimising kurtosis. Can I prove solution uniqueness under particular assumptions using real algebraic geometry or an alternative approach?

I consider a weighted sum of $n$ correlated and identically-distributed random variables. The weights in the sum, $w_i$ for $i=1, 2,...,n$, are non-negative and sum to 1. I am investigating solutions ...
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Irreducible components of real algebraic sets

Let $\mathbb{R}[x_1, \ldots , x_n]$ denote the commutative ring of all polynomials in $n$ variables $x_1, \ldots, x_n$ with coefficients in $\mathbb{R}$. Given a set with $k$ polynomials $\{f_1, . . . ...
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2 votes
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Recommendations for Real Algebraic Geometry and Optimization

I just started looking into positive polynomials on compact semi-algebraic sets and it requires a mixture of optimization, functional analysis and real algebraic geometry. I would like to know if you ...
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2 votes
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Real algebraic set containing a sphere

Consider an irreducible polynomial $P \in \mathbb{R}[x_1,\ldots ,x_n]$ and define $$V := P^{-1}(0) = \left\{ (x_1, \ldots ,x_n) \in \mathbb{R}^n \mid P(x_1,\ldots ,x_n)=0 \right\}$$ It's well known ...
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Upper bound for amount of intervals in intersection of interval sets

I have two sets of numbers which are unions of disjoint intervals, and I have to find an upper bound for how many of such intervals can there be in the intersection of the two sets. Here's a diagram ...
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8 votes
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Does a sequence of $d$-SOS polynomials converge to a polynomial that is also $d$-SOS?

Let $\mathbb{R}[X]_{\leq 2d}$ denote the real vector space of polynomials of degree at most $2d$ in the coordinate ring $\mathbb{R}[X]$ of variety $X$. Definition: A polynomial $f$ is $d$-SOS if ...
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4 votes
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The quotient scheme $X/\Gamma$ when $X$ is separated and every orbit is contained in an affine.

I am trying to solve Problem II.4.7(a) of Hartshorne: The only candidate I can think of for $X_0$ would be the quotient scheme $X/\sigma$. If it exists, it must be unique by the usual argument. First ...
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When does the (real) zero locus $F(x,y,t) = 0$ "look the same" for all values of the parameter $t$?

I have a problem where I would like to know when the zero locus of a deformed polynomial "looks like" (by "looks like" I mean homeomorphic but I am not sure if that is the correct ...
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7 votes
1 answer
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Reference for a real algebraic geometry problem

Disclaimer: I am not a mathematician by training. I encountered the following problem in my research. Assume that I have $N$ real variables $x_1, x_2, \dots, x_N$. I am given $N$ homogeneous ...
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