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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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 ...
Serge the Toaster's user avatar
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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 ...
Fei's user avatar
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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 ...
Super Sanae's user avatar
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 ...
Yromed's user avatar
  • 63
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 ...
mtcli's user avatar
  • 11
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)....
Yromed's user avatar
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1 vote
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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 ...
Yromed's user avatar
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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 ...
Hao Yu's user avatar
  • 185
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 $b(X,Y)=\ell(X)\cdot\ell(Y)$ ...
Anton Petrunin's user avatar
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 $\tilde q$ ...
Anton Petrunin's user avatar
2 votes
1 answer
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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 ...
Ian Gershon Teixeira's user avatar
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 ...
მამუკა ჯიბლაძე's user avatar
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 ...
user505117's user avatar
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 ...
Iosif Pinelis's user avatar
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 ...
Iosif Pinelis's user avatar
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)...
Iosif Pinelis's user avatar
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 ...
Pavel Kocourek's user avatar
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 ...
Pavel Kocourek's user avatar
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? ...
Pavel Kocourek's user avatar
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 ...
giulio bullsaver's user avatar
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 ...
user86954's user avatar
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 ...
Vladimir Zolotov's user avatar
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 ...
Jap88's user avatar
  • 431
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 ...
M.K's user avatar
  • 155
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 \...
Andrius Kulikauskas's user avatar
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 ...
David.D's user avatar
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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 ...
Rami's user avatar
  • 2,571
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 ...
Puzzled's user avatar
  • 8,832
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 ...
Thomas Yang's user avatar
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 ...
David E Speyer's user avatar
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 ...
Ben's user avatar
  • 1,010
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,...
Jürgen Böhm's user avatar
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 ...
Arnold Neumaier's user avatar
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 ...
Iosif Pinelis's user avatar
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 ...
bcp's user avatar
  • 165
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 ...
diddy's user avatar
  • 327
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 ...
lchen's user avatar
  • 459
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,...
Donich's user avatar
  • 318
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 ...
Mark Wildon's user avatar
  • 10.8k
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 ...
Puzzled's user avatar
  • 8,832
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?
A beginner mathmatician's user avatar
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 ...
user159888's user avatar
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(...
Dmitrii Korshunov's user avatar
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 ...
H A Helfgott's user avatar
  • 19.3k
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 ...
Ian Agol's user avatar
  • 66.6k
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 ...
user472602's user avatar
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 - ...
Malkoun's user avatar
  • 4,981
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 ...
Sprotte's user avatar
  • 1,045
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 ...
Iosif Pinelis's user avatar
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) \...
opti's user avatar
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