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 ...
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Are continuous rational functions arc-analytic?
Let $X\subseteq\mathbb{R}^n$ be a smooth semi-algebraic set (for simplicity we can assume $X=B(0,r)$ is a small ball around the origin). A function $f:X\rightarrow \mathbb{R}$ is called a continuous ...
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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 ...
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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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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 ...
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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 ...
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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 ...
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Approximating zero sets of real polynomials with "less complicated" polynomials
Let $K \subset \mathbb{R}^n$ be a compact subset, and let $P(x_1,\dots,x_n)$ be a real multivariable polynomial of degree $d$, whose vanishing set we denote by $Z_P$. Is it plausible to approximate $...
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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)....
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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 ...
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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 ...
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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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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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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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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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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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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 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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Every real variety contains non-singular points
I am looking for a relatively "elementary" proof that every variety in ${\mathbb R}^n$ contains at least one non-singular point.
So far I only have such a proof for the case of hypersurfaces. ...
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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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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,$$
where $f_1,\dots,f_p$ are real polynomials and $x_1,\dots,x_m$ are real independent variables. I'm interested in an upper bound ...
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Roots of linear combination of $x \sin x$
Let $\theta=(\theta_1,\theta_2,\cdots \theta_n)$, and $a_{ij}$ are constants. There is no condition on the positiveness of $a_{ij}$.
Under which condition on $\theta$, such that the following function ...
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What do the Pauli matrices say about the Threefold Way?
The Pauli matrices
$$\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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Intersections of algebraic surfaces with hypercubes of a $d$-dimensional grid
This is a follow-up question, to a question I asked earlier.
See Algebraic curve intersecting square-grid.
Consider $n^d$ unit hypercubes in $d$-dimensional Euclidean space
tightly packed in the ...
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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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On strict positivity and Schmüdgen's Positivstellensatz
Schmüdgen's Positivstellensatz requires the polynomial to be strictly positive on a semialgebraic set. While trying to understand it, I am just wondering if the strictly positive condition can be ...
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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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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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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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Rigid non-archimedean real closed fields
Question. Is there a countable rigid non-Archimedean real closed field?
Background:
As usual, a structure is said to be rigid if the only automorphism of the structure is the identity map.
It is ...
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Real spectrum of ring of continuous semialgebraic functions
Let R be a real closed field, and let U be a semialgebraic subset of $R^n$. Let $S^0(U)$ be the ring of continuous R-valued semialgebraic functions. Also let $\tilde{U}$ be the subset of Spec$_r (R[...
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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 ...
4
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2
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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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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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Measure on real Grassmannians
OK, so I'm reading about this nice measure you can define on a (real) Grassmannian on Wikipedia. Basically, and to save you the trip through the link, consider the Haar measure $\theta$ on $O(n)$, fix ...
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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 ...
38
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6
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Are all polynomial inequalities deducible from the trivial inequality?
I remember learning some years ago of a theorem to the effect that if a polynomial $p(x_1, ... x_n)$ with real coefficients is non-negative on $\mathbb{R}^n$, then it is a sum of squares of ...
3
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1
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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 ...
3
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1
answer
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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 ...