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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Monadic second-order theories of the reals
I’m looking for a survey of monadic second-order theories of the reals.
I’m starting from a 1985 survey by Gurevich which says (p 505) that true arithmetic can be reduced to “the monadic theory of ...
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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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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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Any real algebraic variety is diffeomorphic to a real algebraic variety defined over $\mathbb{Q}$
Given a smooth proper real algebraic variety can you find a smooth proper real algebraic variety defined over $\mathbb{Q}$ that is diffeomorphic to it?
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Effective algorithm to test positivity
Let $f(x_1,\ldots, x_n)$ be a real polynomial in several variables. Is there an effective algorithm to test whether $f$ is positive (or nonnegative) on the whole of ${\mathbb{R}}^n$?
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Is the image of the map $A \to \bigwedge^k A$ closed over $\mathbb{R}$?
Let $V$ a real vector space of dimension $d$. Let $1<k < d-1$. Consider the map induced by the exterior algebra functor:
$$ \psi:\text{End}(V) \to \text{End}(\bigwedge^kV) \, \, \, \, , \, \, ...
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Maximal number of connected components of complement to an affine plane real algebraic curve
Let $X$ be a (singular, reducible) affine plane real algebraic curve of degree $d$.
How we can estimate maximal number of connected components of it's complement in $R^2$ in terms of degree?
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Exact arithmetic for real algebraic numbers
There was a reply to a question (that I can't find) which mentioned SARAG (Some Algorithms
in Real Algebraic Geometry) see http://perso.univ-rennes1.fr/marie-francoise.roy/bpr-ed2-posted2.html. This ...
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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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Cohomology of Structure Sheaves: Algebraic, Constructible and more
I am not an algebraic geometer, but I am a topologist who uses sheaves. I have studied some algebraic geometry and am interested in what happens as I reduce the amount of rigidity in the structure ...
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Knots realized as algebraic curves
Two questions:
Q1. Have researchers worked out minimum-degree
real algebraic curves in $\mathbb{R}^3$ realizing specific knots?
Some work on the trefoil is reported in this MSE question.
&...
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"Real algebraic varieties" vs finite type separated reduced $\mathbb{R}$-schemes with dense $\mathbb{R}$-points
This question is partly motivated by a few comments here. Let me denote by $R$ the (real-closed) field of real numbers $\mathbb{R}$; everything is probably the same over an arbitrary real-closed field....
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An explicit reconstruction of a matrix from its minors
$\newcommand{\End}{\operatorname{End}}$
$\newcommand{\GL}{\operatorname{GL}}$
$\newcommand{\Cof}{\operatorname{cof}}$
Let $V$ be a $d$-dimensional real vector space. ($d \ge 4$). Fix an odd integer $...
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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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Nonnegativity conditions for a polynomial in two variables?
Let
$$P(X,Y)= c_{22}X^2Y^2 +c_{21}X^2Y +c_{12}XY^2 +c_{20}X^2 +c_{11}XY+c_{02}Y^2+c_{10}X+c_{01}Y+c_{00}$$
be a polynomial of two variables $X$ and $Y$ with real coefficients $c_{ij}$.
What are the ...
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Embeddings and triangulations of real analytic varieties
This is a follow up question to my answer here How do you define the Euler Characteristic of a scheme?
A real analytic space is a ringed space locally isomorphic to $(X,O/I)$ where $X$ is the zero ...
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A question on symmetric matrices
$\newcommand{\R}{\mathbb{R}}$
The question is
Is there a constructive (say, parametric) description of the set (say $M_n$) of all symmetric matrices $A\in\R^{n\times n}$ such that all the diagonal ...
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Solving system of bilinear equations
Consider a collection of $m$ matrices $A_i$ of size $n\times n$, and a vector $b$ of size $m$. I want to solve the bilinear system
$$\left\{ x^T A_i y = b_i : i = 1,\dots,m \right\}$$
in variables $x,...
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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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What is the topology on the set of field orders
Inspired by this question I was wondering whether there is a natural topology on the set of all orders on a field (that extend a given order on a subfield)?
For example for the function field $\...
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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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Connected components of the complement of a degree-d affine hypersurface
Let $n$ and $d$ be positive integers, and $f\in\mathbb{R}[x_1,\dots,x_n]$ be a polynomial of degree $d$. Let's consider the zero-set $M = \{x \in \mathbb{R}^n: f(x) = 0\}$ of $f$.
Can we estimate ...
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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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The Universality Theorem by Mnev for uniform oriented matroids of rank 4 and higher
According to the Universality Theorem by Mnev (see below theorem 8.6.6 from [1]), for any open semialgebraic variety V there is a uniform oriented matroid of rank 3 whose realization space is stably ...
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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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Locally Closed Orbits in Real Algebraic Geometry
Let $G$ be a real algebraic group, and let $X$ be a real affine $G$-variety. I am looking for conditions on $G$ and $X$ for which the $G$-orbits are known to be locally closed in the Zariski topology ...
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Polynomials (or analytic functions) vanishing on a real algebraic set
I have seen the following result stated several times in the literature, without proof:
Let $\mathbb{K}$ be $\mathbb{R}$ or $\mathbb{C}$, and assume $P\in\mathbb{K}[X_{1},\ldots,X_{n}]$ is an ...
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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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Connectedness of semialgebraic sets via CAD
I do not know whether there is a standard or some traditional ways to decide whether a semialgebraic set is connected or not.
One way I know is the cylindrical algebraic decomposition (CAD) algorithm. ...
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Differential operators that preserve real-rootedness
Is there some description of polynomial differential operators, $\mathcal{D}=\sum f_i(x) D_x^i$ such that, if $h$ is a polynomial all of whose roots are in $[0,1]$, then so are all the roots of $\...
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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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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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Cylindrical Decomposition vs Morse decomposition
Suppose I have a polynomial Morse function $f: \mathbb{R}^n \to \mathbb{R}$. Consider the ideal $I(\nabla f)$ generated by the partial derivatives $\partial_i f$, and assume that the real zero-set of ...
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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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One of Poincaré's theorems about positive rational functions
A rational function in $ \mathbb{R}[x_1, x_2] $ is called positive if $f = g/h$ with $g,h \in \mathbb{R}_{\geq 0}[x_1, x_2]$. Are there some references about the following theorem given by Poincare?
...
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Can one push a path out of an algebraic set with control on length?
Consider a real algebraic set $A\subseteq\mathbb{R}^n$. The case I am interested in is a homogeneous cone (i.e., if $x\in A$ then also $\lambda x\in A$ for each scalar $\lambda>0$) where each ...
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Algebraic curve intersecting square-grid
Let us subdivide the unit square into square-grid cells with sidelength $w$. This will give us roughly $w^{-2}$ cells.
Formally
$$ g_{ij} = \{(wi, wj) + (x,y) : 0\leq x,y\leq w \},$$
for $i,j = 0,\...
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Approximation of curves
When constructing minimax (sup-norm) polynomial approximations of real-valued functions, well-known results say (roughly speaking) that optimal solutions are characterized by the fact that they have ...
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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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Chebyshev-like Problem for Plucker Coordinates
$\newcommand{\R}{\mathbb{R}}\newcommand{\D}[1]{\Delta_{#1}}\newcommand{\set}[1]{\{#1\}}\newcommand{\abs}[1]{\lvert#1\rvert}$
Let $n=2d+1$ be an odd integer, let $Gr(2,n)$ denote the Grassmmanian over $...
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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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Non-negative polynomials $f(p), p\in P$ from Polynomial ideal where $P$ compact polytope?
Partially related to Non-negative Polynomials from Polynomial Ideal? where focus on graph theory but now I want to understand more generally how to determine non-negativity in more general case.
A. ...
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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 ...
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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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Non-negative Polynomials from Polynomial Ideal?
Related to the thread Nonnegativity conditions for a polynomial in two variables? but on more than two variables.
Suppose a probability vector $p$ belongs to a compact polytope where for each entry $...