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).
312
questions
0
votes
0
answers
123
views
Noether's formula for real algebraic surfaces
Is there a version of Noether's formula for the Euler characteristic of a surface for Real algebraic surfaces?
Specifically, given $X$ a real algebraic compact smooth surface, what is the relationship ...
0
votes
0
answers
103
views
How many isolated points can a degree $d$ planar curve have?
Let $p(x,y)\in\mathbb R[x,y]$ be a bivariate polynomial of degree $d$. What is the maximum possible number of its acnodes (i.e. isolated roots in $\mathbb R^2$ not counting multiplicities)?
A pretty ...
7
votes
1
answer
254
views
How to construct such a real algebraic curve
Suppose $L$ is a real line in $\mathbb{RP}^2$, my question is: for a given postive integer $n$, is it possible to find a real projective algebraic curve $C$ of degree $n$ with maximum connected ...
3
votes
1
answer
201
views
Toric varieties as hypersurfaces of degree (1, ..., 1) in a product of projective spaces
I wonder if some hypersurfaces of multi-degree $(1, ..., 1)$ in a product of projective spaces are toric, with polytope a union of polytopes of toric divisors of the ambient space. This question is ...
1
vote
1
answer
60
views
Number of intersection points of a system of quartic polynomials induced by the projected matrix commutator
Let $A, B \in \mathbb{R}^{d \times d}$ denote two symmetric positive definite matrices. I am interested in solutions $V_r \in \mathbb{R}^{d \times r}, 1 < r < d$ to the system of quartic ...
0
votes
0
answers
109
views
Embedding of the first Hirzebruch surface in $\mathbb{P}^4$ as a cubic surface
The first Hirzebruch surface (the blow-up of $\mathbb{P}^2$ at one point) is a projective toric surface that naturally embeds into $\mathbb{P}^4$ as a cubic surface (sometimes called the cubic scroll)....
1
vote
0
answers
60
views
Embedding toric varieties in other toric varieties as a real algebraic hypersurface
In the question On a Hirzebruch surface, I've seen that the $n$-th Hirzebruch surface is isomorphic to a surface of bidegree $(n,1)$ in $\mathbb{P}^1\times \mathbb{P}^2$. I am trying to answer the ...
8
votes
1
answer
594
views
Connеcted components of irreducible algebraic varieties
I am wondering what is the possible (or maximum) number of connected components for an irreducible algebraic variety in $\mathbb R^n$ defined by a degree $d$ polynomial (i.e. hypersurface) in $\mathbb ...
12
votes
1
answer
884
views
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 $b(X,Y)=\ell(X)\cdot\ell(Y)$ ...
15
votes
2
answers
1k
views
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 $\tilde q$ ...
2
votes
1
answer
89
views
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 ...
2
votes
0
answers
96
views
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 ...
6
votes
1
answer
214
views
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 ...
5
votes
2
answers
294
views
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 ...
6
votes
1
answer
209
views
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 ...
4
votes
1
answer
447
views
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
$$\sum_{i,j}(G^5)...
2
votes
1
answer
140
views
How many strict local minima can a quintic polynomial in two real variables have?
A quadratic or cubic polynomial (in two variables) can have at most one strict local minimum. A quartic polynomial can have up to five strict local minima [1]. So, how many strict local minima can a ...
8
votes
1
answer
640
views
How many saddle points can a quartic polynomial in two real variables have? All 9?
By Bézout's theorem a quartic polynomial $p(x,y)$ can have at most 9 isolated critical points. Can all of them be saddle points?
In case of a cubic polynomial there is a mechanical way to answer this ...
7
votes
1
answer
984
views
Can a cubic polynomial in two real variables have three saddle points?
Is there a cubic polynomial $c(x,y)$ with exactly 3 saddle point critical points?
In other words, can a cubic polynomial in two variables have three critical points, all of which are saddle points? ...
0
votes
0
answers
66
views
Feasibility of a polynomial system of equalities and inequalities
Consider a system of the form $f_i(x) = 0$ and $g_j(x) \ge 0$ ,where $f_i,i=1,\dots,r$ and $g_j,j=1,\dots,s$ are polynomials in real unknowns $x_i,i=1,\dots,n$ with rational coefficients.
Is there a ...
3
votes
2
answers
407
views
The real dimension of any real algebraic set equals the complex dimension of its complexification
I want to prove the following statement. Please help!
Given any semialgebraic set $A$, consider its real Zariski closure $V_{\mathbb{R}}$ (which always has the same real dimension of $A$). Now ...
1
vote
0
answers
195
views
Does Thom-Milnor theorem provide a bound for the number of connected components of a zero set of a polynomial system?
I have a system $$f_1(x_1,\dots,x_m) = 0,...,f_p(x_1,\dots,x_m) = 0,$$
where $f_1,\dots,f_p$ are real polynomials and $x_1,\dots,x_m$ are real independent variables. I'm interested in an upper bound ...
29
votes
1
answer
1k
views
Can a real quartic polynomial in two variables have more than 4 isolated local minima?
This question: "Can a real quartic polynomial in two variables have at most 4 isolated local minima?" came up in this post on Math SE but with no answer so far.
Finding examples of 4 ...
0
votes
1
answer
279
views
Roots of linear combination of $x \sin x$
Let $\theta=(\theta_1,\theta_2,\cdots \theta_n)$, and $a_{ij}$ are constants. There is no condition on the positiveness of $a_{ij}$.
Under which condition on $\theta$, such that the following function ...
2
votes
1
answer
212
views
What do the Pauli matrices say about the Threefold Way?
The Pauli matrices
$$\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 \...
6
votes
2
answers
627
views
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 ...
4
votes
1
answer
191
views
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 ...
2
votes
0
answers
188
views
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 ...
3
votes
0
answers
74
views
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 ...
10
votes
2
answers
433
views
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 ...
2
votes
1
answer
164
views
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 ...
3
votes
0
answers
136
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
142
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
439
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 ...
9
votes
0
answers
284
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 ...
0
votes
1
answer
613
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 ...
1
vote
1
answer
304
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 ...
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
280
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 ...
3
votes
1
answer
188
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 ...
5
votes
4
answers
353
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
214
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
234
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(...
4
votes
2
answers
273
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 ...
3
votes
1
answer
204
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 ...
1
vote
0
answers
239
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
114
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 - ...
1
vote
0
answers
94
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 ...
2
votes
2
answers
205
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
165
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) \...