Questions tagged [real-algebraic-geometry]

Real algebraic geometry is the study of real solutions to algebraic equations with real coefficients. Its methods are rather different from classical algebraic geometry, which is typically done over an algebraically closed field (like the complex numbers).

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3 votes
0 answers
63 views

Topology types in families of real or complex varieties

In René Thom, "Structural Stability and Morphogenesis" on p. 21ff there is the following statement: Let $$P_j(x_i,s_k) = 0$$ be a set of polynomial equations over the real or complex numbers,...
0 votes
0 answers
82 views

Multivariate polynomials with given real zeros

Given natural numbers $N$ and $n$, I am looking for a family $P_{Nn}$ of polynomials in $n$ variables with the following properties: (i) Every polynomial in $P_{Nn}$ is determined by the function ...
4 votes
2 answers
393 views

About Euclidean distances

$\newcommand\R{\mathbb R}$Let $0<d_1<\cdots<d_k<\infty$ and let $m_1,\dots,m_k$ be any integers $\ge1$. Let $n:=m_1+\dots+m_k-1$. Let $d$ denote the Euclidean distance in $\R^n$. Do then ...
1 vote
0 answers
71 views

Computer algebra tools for finding real dimension of an algebraic variety

I have a system of polynomial equations with the unknowns being real numbers. The set of solutions is infinite. What software can I use to compute the real dimension of the solution set? The CAD-based ...
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0 votes
1 answer
43 views

Plot two implicit surfaces in 3D and highlight their intersection [closed]

I want to plot the two surfaces which are defined in $ \mathbb{ R }^3 \ni ( x, y, z ) $ via the equations $ 0 = y^2 - x*(x^2 + 1) $ and $ 0 = z^2 - y*(y^2 + 1) $, respectively. Moreover, I want also ...
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1 vote
1 answer
278 views

An inequality in four variables

Let $f(x,y)=\frac{10xy-(x+y)+1}{8xy-2(x+y)+5}$ and $g(x,y)=\frac{1}{4}\left[1+\frac{1}{3}(4x-1)(4y-1)\right]$. I want to prove that for any $0.5\le a\le b\le 1$ and $0.7\le c\le d\le 1$, it holds that ...
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16 votes
1 answer
1k views

Is the "equidistant curve" to an algebraic curve algebraic?

Let $ L \subseteq \mathbb{R}^2 $ be a smooth real algebraic curve. Let's fix some parameter $ \delta \in \mathbb{R} $ and for every point $ (x,y) \in L $ define $$ L_{\delta}(x,y) = (x,y) + \delta n(x,...
10 votes
1 answer
246 views

Rational even polynomials maximally tangent to the unit circle

This question is motivated by College Mathematics Journal problem 1196, proposed by Ferenc Beleznay and Daniel Hwang. My solution to this problem (pre-publication version here) uses Chebyshev ...
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4 votes
1 answer
162 views

Positivity of real functions in two variables

Assume that $f_0,f_1,f_2$ are polynomial functions of degree two in two variables. This means that the $f_i$ are linear combinations with real coefficients of $x^2,xy,x,y^2,y,1$. Consider the function ...
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5 votes
4 answers
309 views

Dual norm of a subspace of $\ell_\infty^3$

We define a norm on $\mathbb C^2$ as $\|(\alpha,\beta)\|:=\max\left\{|\alpha|,|\beta|,\big|\frac{\alpha+\beta}{\sqrt{2}}\big|\right\}.$ Can the dual norm be calculated explicitly?
6 votes
1 answer
190 views

Fixed points of a function $z\mapsto\overline{P(z)}$ of a complex variable

The equation $z^2=\overline{z}$ has four zeros and this example motivates us to generalize the problem to this form; How many zeros does the equation $P(z)=\overline{z}$ have if $P(z)$ is a polynomial ...
8 votes
1 answer
191 views

Projections of compact real algebraic sets

Suppose that $M$ is a compact, real algebraic subset of $\mathbb R^n$ and $f:\mathbb R^n \to \mathbb R^m$ is the projection to the first $m$ coordinates. If $f$ maps $M$ bijectively unto its image $f(...
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4 votes
2 answers
264 views

Quantifier elimination in $S^1$

Does quantifier elimination (by cylindrical decomposition) work for systems of polynomial equations and inequalities where some or all of the variables are complex numbers of unit modulus, rather than ...
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3 votes
1 answer
177 views

Reference request: ordered list of dimensions of components of a variety?

Let $V$ be an affine real algebraic set. That is, $V$ is the zero set of some polynomials in $\mathbb{R}^n$. I would like to show that there is not a proper algebraic subset $W\subset V$ which admits ...
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1 vote
0 answers
229 views

When is a topological manifold which is not an almost complex manifold algebraic?

When is a topological manifold which is not an almost complex manifold isomorphic to a real algebraic variety in the sense of locally ringed space? It is well known that Serre's GAGA theorem solves ...
2 votes
0 answers
100 views

An approach to the Atiyah problem on configurations via real semi-algebraic geometry

The cross-ratio $$ C(z_1, z_2; z_3, z_4) = \frac{(z_4 - z_1)(z_3 - z_2)}{(z_3 - z_1)(z_4 - z_2)} $$ has a degree $3$ analogue $$ H(z_1, z_2, z_3; z_4, z_5, z_6) = \frac{(z_5 - z_1)(z_6 - z_2)(z_4 - ...
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1 vote
0 answers
87 views

Finite number of topological spaces realized by varieties of bounded degree?

I am not familiar with algebraic geometry so I am sorry if this question is terribly ignorant. Any basic reference is appreciated. Is there a finite bound on the number of topological spaces that can ...
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2 votes
2 answers
138 views

On an angle distribution of a random linear subspace of a given dimension

$\newcommand\R{\mathbb R}$ Let $u$ be a fixed unit vector in $\R^n$, and let $\Pi_u$ be the hyperplane in $\R^n$ with normal vector $u$. Let $B$ be the (say open) unit ball in $\R^n$ centered at the ...
2 votes
0 answers
145 views

Certificates of connectivity of basic semi-algebraic sets

Given real polynomials $p_1, \ldots, p_n \in {\mathbb R}[x_1, \ldots, x_d]$, consider the closed basic semi-algebraic set $S \subseteq {\mathbb R}^d$ given by $$S := \{x \in {\mathbb R}^d : p_i(x) \...
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6 votes
1 answer
373 views

Homotopy equivalence of stably equivalent semialgebraic sets

In his book [1], Richter-Gebert introduces a notion of stable equivalence for primary basic semialgebraic sets (subsets of $\mathbb{R}^n$ defined by a conjunction of polynomial equations and strict ...
1 vote
0 answers
94 views

Only Zariski-closed subsets of compact Lie groups with nonempty interior have nonzero measure

In this question, the following fact was used by the respondent A Zariski-closed subset of a compact Lie group $G$ with nonzero Haar measure contains a coset of $G^0$, the connected component of $G$ ...
5 votes
1 answer
153 views

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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2 votes
1 answer
229 views

If a variety over a real closed field has finitely many points they are singular

Let $F$ be a real closed field. Let $X$ be a positive-dimensional algebraic variety over $F$. If $X$ finitely many $F$-points are they all singular?
4 votes
0 answers
106 views

Delta distributions that are smooth on strata of a singular manifold

This is a mild reformulation of a previous question. Let $R = C^\infty(\mathbb{R}^N)$ and let $I$ be an ideal in $R$ which cuts out an $n$-dimensional "singular $C^\infty$ manifold $X$" in $\...
6 votes
0 answers
78 views

forms on singular spaces that can be integrated on an LCI

I'd like to characterize rational $k$-forms on a singular scheme $X$ which can be integrated on any (real) bounded submanifold that is locally LCI in $X$ in a real sense (i.e., which is the real ...
2 votes
0 answers
205 views

Characterizing non-zero polynomials on semialgebraic sets: a kind of positivstellensatz generalization

A polynomial positivstellensatz is an algebraic characterization of polynomials which are positive on a semialgebraic sets. Is there a similar kind of characterization which can determine whether a ...
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8 votes
2 answers
638 views

Bialynicki-Birula decomposition for real analytic varieties

Let $X$ be a smooth complex algebraic variety endowed with a $\mathbb{C}^*$ action. We assume also to have an antiholomorphic involution $\sigma$ over $X$ such that it anticommutes with the action ...
5 votes
2 answers
352 views

Invariant theory over $\mathbb R$

$\DeclareMathOperator\SO{SO}$Suppose we have a (continuous) linear action of $\SO(n,\mathbb R)$ on a vector space $\mathbb R^N$. Consider the ring of invariants $A\subset \mathbb R[x_1,\ldots, x_N]$, ...
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4 votes
1 answer
104 views

Classes of curves closed under Minkowsky sum

Let $C$ be a class of plane curves, regarded as subsets of $\mathbb{R}^2$ (parametrization won't matter), I'm thinking for example of splines or algebraic subsets. Let $D$ be a class of topological ...
2 votes
0 answers
175 views

Do singularities in real algebraic varieties have measure zero?

This question is related to Do proper Zariski closed sets of algebraic sets have measure zero For algebraic varieties on complex numbers, it is easy to see that the locus of their singularies are set ...
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5 votes
2 answers
302 views

Nowhere negative polynomials form a semialgebraic set

Let $P_{d, n}$ be the space of polynomial maps $\mathbb{R}^n\to \mathbb{R}$ of degree at most $d$. Is the subset $S\subset P_{d, n}$ of nowhere negative polynomials semialgebraic?
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6 votes
1 answer
272 views

Real analyticity of continuous function via restriction to analytic curves

Suppose $X\subset \mathbb R^n$ is an irreducible real analytic sub-variety (i.e. the set of solutions of a system $f_1=\ldots=f_k=0$ with $f_i$ analytic) Let $x\in X$ be a point and let $F: X\to \...
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4 votes
1 answer
299 views

Is a complex or real algebraic variety homotopically equivalent to a CW complex?

Let $k$ be either the field $\Bbb C$ of complex numbers or the field $\Bbb R$ of real numbers. Let $X$ be an algebraic variety over $k$, say, quasi-projective and smooth (but not necessarily ...
4 votes
0 answers
249 views

Algebraic properties of geodesics

This is a question related to my last post. I will use the same definition here. A complete smooth manifold $M$ with an affine connection $\nabla$ is said to have an algebraic g-model of dimension $n$ ...
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7 votes
0 answers
576 views

Geodesics on algebraic manifold

A nonsingular algebraic manifold is an immersed manifold (slightly different from the usual embedded algebraic manifold) $M \subseteq \Bbb{R}^n$ that is also a nonsingular algebraic set (which means $...
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5 votes
1 answer
955 views

An almost complex structure on the real $n$-sphere $S^n$

If $R\mathrel{:=}\mathbb{R}[x_1,\dotsc,x_{n+1}]/(x_1^2+\dotsb+x_{n+1}^2-1)$ and $S^n\mathrel{:=}\operatorname{Spec}(R)$ is the real $n$-spere, a classical result of Borel and Serre says that the only ...
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2 votes
0 answers
114 views

Degree of polynomials describing projection of algebraic set

Consider an algebraic subset $V\subseteq \mathbb{R}^{n+1}$ defined as the zero set of polynomials ${f_i}$ and the projection map $\pi: \mathbb{R}^{n+1}\to \mathbb{R}^n$ deleting the last entry. By the ...
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2 votes
0 answers
60 views

Uniform Łojasiewicz constant in 2D

Łojasiewicz inequality is a classical result in real algebraic geometry. In particular, for any given polynomial $f:\mathbb R^2\to \mathbb R$ there is some $C>0$ and some $\alpha>0$ such that ...
1 vote
2 answers
632 views

Rational bijections $\mathbb R\to(0;1)$ [closed]

Notation: $$ (0;1)\ :=\ \{x\in\mathbb R:\ 0<x<1\}$$ There are simple rational stretches $\ f \colon (0; \, 1)\to\mathbb R,\ $ e.g. let $\ s\in(0;\, 1);\ $ then $$ f(x)\ :=\ \frac{1-s}{1-x}-\...
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1 vote
0 answers
56 views

Lower bounds on lengths of sum-of-squares representations of particular polynomials

I am looking for literature on the problem of finding minimal (in the sense of number of terms) sum-of-squares representations of particular non-negative multivariate polynomials with rational ...
2 votes
0 answers
93 views

Complexity of polynomial inequalities

What is known about the complexity of deciding whether a finite set of polynomial inequalities in $n$ real variables with integer coefficients is satisfiable? Decidability is guaranteed by Tarski's ...
1 vote
1 answer
151 views

Product of subgroups of $SU(8)$ algebraic set?

Consider the special unitary group SU(8) acting on $\mathbb{C}^8\stackrel{\sim}{=}(\mathbb{C}^2)^{\otimes 3}$. In particular, I am interested in the two subgroups $G_1=\mathrm{id}_{\mathbb{C}^2}\...
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8 votes
2 answers
339 views

Antiderivative of totally real polynomials

Let us say that a polynomial with real coefficients is totally real if all its complex roots are real and distinct. Let $P \in \Bbb R [X]$ be totally real. Is it true that $$Q(X)=\int_0^XP(t)\,dt+aP(X)...
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6 votes
1 answer
142 views

Subsets of a ball/sphere with the largest sum of distances

$\newcommand\R{\mathbb R}\newcommand\S{\mathbb S}$Let $B_d$ and $S_{d-1}$ denote, respectively, the closed unit ball and the unit sphere in $\R^d$. Let us say that a finite subset $F$ of $B_d$ is ...
1 vote
1 answer
158 views

Can I prove that a polynomial representing the 4th moment of a weighted-sum of random variables is a sos?

I am looking at the 4th central moment of a weighted-sum of correlated random variables, which takes the form $$\mu_4 = \sum_{i,j,k,l=1}^n w_i w_j w_k w_l \mu_{ijkl}$$ where $\mu_{ijkl}$ are the ...
  • 173
11 votes
0 answers
323 views

Is there yet an example of a non-negative convex polynomial that cannot be written as a sum-of-squares?

I have read that it remains an open question, whether an example can be constructed of a non-negative convex polynomial that cannot be written as a sum-of-squares. My reading includes the following ...
  • 173
6 votes
1 answer
180 views

Constructing M-curves à la Hilbert

I have been reading some text about Harnack's theorem. The theorem basically says that for degree $d$, the maximal number of connected components in the real (projective) plane of a plane curve with ...
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6 votes
1 answer
156 views

Decomposition of real algebraic varieties into manifolds

I apologize in advance if this question is too elementary for MO. I am new to the field of algebraic geometry. I am dealing with a (real) algebraic variety $V$ of (Krull) dimension $n$. I keep reading ...
  • 417
1 vote
0 answers
54 views

Minimising kurtosis (non-convex). Can I use algebraic geometry or alternate methods to show uniqueness of a particular solution?

I consider a weighted sum of $n$ identically-distributed correlated random variables. The weights in the sum, $w_i$ for $i=1, 2,...,n$, satisfy $w_i>=0$ and $\sum_{i=1}^{n}w_i=1$. I am ...
  • 173
3 votes
0 answers
57 views

The dimension of the $k$-independence

$\newcommand\Om\Omega$Let $(\Om,F,P)$ be a probability space. For some natural $n$, let $A_1,\dots,A_n$ be events, that is, members of the $\sigma$-algebra $F$. For $k\in[n]:=\{1,\dots,n\}$, these ...

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