# 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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### Positive 4-form

Denote by $W$ the space of all symmetric bilinear forms on $\mathbb{R}^n$.
Let $Q$ be a quadratic form on $W$.
Suppose that $Q(b)\geqslant 0$ for any $b\in W$ such that $b(X,Y)=\ell(X)\cdot\ell(Y)$ ...

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### Positive quadratic polynomial

Let $S$ be solutions of a system of quadratic polynomials on $\mathbb{R}^n$.
Suppose $q$ is another quadratic polynomial such that $q|_S\geqslant 0$.
Is it possible to find a polynomial $\tilde q$ ...

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### Sufficient condition for pair of real quadrics to have real intersection

In the following, when I talk about the zero of a homogeneous polynomial I always mean a projective zero.
Let $ q $ be a real quadric. Then $ q $ has a real zero if and only if $ q $ has indefinite ...

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### Real-isability of a (relatively small) subconfiguration of the Klein configuration

The Klein configuration consists of $60$ points and $60$ planes in $\mathbb C\mathbf P^3$, each point lying on $15$ of the planes and each plane containing $15$ of the points. It appears, among many ...

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### How small need a perturbation be to not change the diffeomorphism type of a variety?

Let $f,g \in \mathbb{R}[x_0,\dots,x_k]$ be homogeneous polynomials and $X:=Z(f) \subset \mathbb{RP}^k$ be the projective variety defined by $f$.
Assume that $X$ is smooth and has codimension $1$.
Then ...

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### An inequality problem for certain positive-definite matrices

Suppose that $G=(G_{ij})$ is an $n\times n$ positive-definite symmetric matrix with the diagonal entries all equal $1$ and all off-diagonal entries $<0$. Let $a$ be a column $n\times1$ matrix such ...

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### An inequality for certain positive-definite matrices

Suppose that $G=(G_{ij})$ is an $n\times n$ positive-definite symmetric matrix with the diagonal entries all equal $1$ and all off-diagonal entries $\le0$. Let $a$ be a column $n\times1$ matrix with ...

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### An inequality for certain positive-semidefinite matrices

Suppose that $G=(G_{ij})$ is a positive-semidefinite symmetric matrix with the diagonal entries all equal $1$ and all off-diagonal entries $\le0$. Does it then necessarily follow that
$$\sum_{i,j}(G^5)...

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### How many strict local minima can a quintic polynomial in two real variables have?

A quadratic or cubic polynomial (in two variables) can have at most one strict local minimum. A quartic polynomial can have up to five strict local minima [1]. So, how many strict local minima can a ...

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### How many saddle points can a quartic polynomial in two real variables have? All 9?

By Bézout's theorem a quartic polynomial $p(x,y)$ can have at most 9 isolated critical points. Can all of them be saddle points?
In case of a cubic polynomial there is a mechanical way to answer this ...

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### Can a cubic polynomial in two real variables have three saddle points?

Is there a cubic polynomial $c(x,y)$ with exactly 3 saddle point critical points?
In other words, can a cubic polynomial in two variables have three critical points, all of which are saddle points? ...

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### Feasibility of a polynomial system of equalities and inequalities

Consider a system of the form $f_i(x) = 0$ and $g_j(x) \ge 0$ ,where $f_i,i=1,\dots,r$ and $g_j,j=1,\dots,s$ are polynomials in real unknowns $x_i,i=1,\dots,n$ with rational coefficients.
Is there a ...

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### The real dimension of any real algebraic set equals the complex dimension of its complexification

I want to prove the following statement. Please help!
Given any semialgebraic set $A$, consider its real Zariski closure $V_{\mathbb{R}}$ (which always has the same real dimension of $A$). Now ...

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### Does Thom-Milnor theorem provide a bound for the number of connected components of a zero set of a polynomial system?

I have a system $$f_1(x_1,\dots,x_m) = 0,...,f_p(x_1,\dots,x_m) = 0,$$
where $f_1,\dots,f_p$ are real polynomials and $x_1,\dots,x_m$ are real independent variables. I'm interested in an upper bound ...

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### Can a real quartic polynomial in two variables have more than 4 isolated local minima?

This question: "Can a real quartic polynomial in two variables have at most 4 isolated local minima?" came up in this post on Math SE but with no answer so far.
Finding examples of 4 ...

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### Roots of linear combination of $x \sin x$

Let $\theta=(\theta_1,\theta_2,\cdots \theta_n)$, and $a_{ij}$ are constants. There is no condition on the positiveness of $a_{ij}$.
Under which condition on $\theta$, such that the following function ...

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1
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### What do the Pauli matrices say about the Threefold Way?

The Pauli matrices
$$\sigma_1=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix},
\sigma_2=\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix},
\sigma_3=\begin{pmatrix} 1 & 0 \\ 0 & -1 \...

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### Consequences of Nash-Tognoli Theorem

The Nash-Tognoli theorem states that every closed and smooth manifold is diffeomorphic to a real algebraic variety. This appears to me as a very strong and surprising fact.
However, I am not aware of ...

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### Is a continuous rational function Lipschitz?

Let $f\in \mathbb{R}(x_1,\ldots,x_n)$ be a rational function. Suppose that $f$ is continuous on $\mathbb{R} ^n$. Must it be Lipschitz on the unit ball?
This question might be related to Are continuous ...

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### Degree four polynomials with no real roots

Consider a degree four polynomial
$$
f = a_4x^4 + a_3x^3 + a_2x^2 + a_1x+ a_0 \in \mathbb{R}[x]
$$
with real coefficients. The discriminant $\Delta_f$ of $f$ is a homogeneous polynomials of degree six ...

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### Monge–Ampère equation with polynomials

Let $P$ be a real polynomial of $n$ variables. Does the Monge–Ampère equation
$$
\det D^2 Q=P
$$
always have a (global) real polynomial solution $Q$?
I think this should be something standard, but I ...

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### A category of spaces in which $\mathbb{R}_{>0}$ is not isomorphic to $\mathbb{R}$

$\def\CC{\mathbb{C}}\def\RR{\mathbb{R}}$I am writing up notes on totally positivity in flag varieties. I often have the following situation: I have a complex variety $X$ defined over $\RR$ and a real ...

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### Is the Segre embedding of two real varieties a real variety?

$\newcommand{\complex}{\mathbb{C}}\newcommand{\real}{\mathbb{R}}\newcommand{\proj}{\mathbb{P}}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Seg{Seg}$I apologize in advance for my naïve ...

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### 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,...

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### 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 ...

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### 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 ...

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### 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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### 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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### 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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### 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,...

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### 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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### 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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### 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?

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### 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 ...

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### 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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### 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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### 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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### 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 ...

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### 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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### 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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### 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 ...

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### 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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### 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 ...

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0
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### 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
$G$ ...

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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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### 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?

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### 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 $\...

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### 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 ...

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### 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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### 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 ...