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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29 votes
1 answer
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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 ...
Jap88's user avatar
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7 votes
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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 ...
Thierry Zell's user avatar
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17 votes
1 answer
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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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13 votes
3 answers
834 views

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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12 votes
1 answer
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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) \, \, \, \, , \, \, ...
Asaf Shachar's user avatar
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8 votes
2 answers
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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?
probably's user avatar
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6 votes
1 answer
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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 ...
Bruce Westbury's user avatar
4 votes
2 answers
2k views

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. ...
Adam Sheffer's user avatar
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22 votes
5 answers
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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 ...
Justin Curry's user avatar
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22 votes
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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.   &...
Joseph O'Rourke's user avatar
18 votes
1 answer
849 views

"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....
Qfwfq's user avatar
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17 votes
1 answer
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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 $...
Asaf Shachar's user avatar
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15 votes
2 answers
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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 $\tilde q$ ...
Anton Petrunin's user avatar
12 votes
1 answer
885 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
11 votes
3 answers
1k views

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 ...
Dan Ismailescu's user avatar
9 votes
2 answers
1k views

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 ...
algori's user avatar
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8 votes
1 answer
437 views

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 ...
Iosif Pinelis's user avatar
7 votes
1 answer
2k views

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,...
grok's user avatar
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7 votes
0 answers
633 views

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 $...
Zerox's user avatar
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7 votes
1 answer
537 views

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 $\...
HenrikRüping's user avatar
7 votes
1 answer
989 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
7 votes
2 answers
1k views

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 ...
Yaroslav Ivanov's user avatar
6 votes
1 answer
210 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
6 votes
1 answer
501 views

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 ...
Jae's user avatar
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6 votes
0 answers
92 views

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 ...
Dmitry Vaintrob's user avatar
5 votes
1 answer
725 views

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 ...
Peter Crooks's user avatar
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5 votes
1 answer
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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 ...
user111's user avatar
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5 votes
2 answers
361 views

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 ...
Anonymous Coward's user avatar
5 votes
1 answer
156 views

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. ...
user91646's user avatar
5 votes
0 answers
259 views

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 $\...
David E Speyer's user avatar
4 votes
1 answer
449 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
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
4 votes
0 answers
119 views

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 ...
Simon Segert's user avatar
3 votes
0 answers
140 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
3 votes
1 answer
310 views

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? ...
Jianrong Li's user avatar
  • 6,101
3 votes
0 answers
114 views

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 ...
Mikhail Katz's user avatar
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3 votes
2 answers
425 views

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,\...
Till's user avatar
  • 469
3 votes
2 answers
720 views

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 ...
bubba's user avatar
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2 votes
2 answers
207 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
132 views

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 $...
Vadim Ogranovich's user avatar
2 votes
1 answer
380 views

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 ...
Till's user avatar
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1 vote
0 answers
46 views

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. ...
hhh's user avatar
  • 143
1 vote
1 answer
252 views

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 ...
Brian's user avatar
  • 173
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
0 votes
0 answers
118 views

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 $...
hhh's user avatar
  • 143