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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5
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1answer
248 views

The first part of the Hilbert sixteenth problem for elliptic polynomials

A polynomial $P(x,y)\in \mathbb{R}[x,y]$ is called an elliptic polynomial if its highest homogeneous part does not vanish on $\mathbb{R}^2\setminus\{0\}$. Inspired by the first part of the Hilbert ...
9
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1answer
195 views

A property of varieties between unirational and retract rational

EDIT: The vague question Q1 below is partially answered, while the concrete question Q2 seems to be still open. Let $V$ be a geometrically integral variety over a field $K$. I consider the following ...
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6answers
1k views

Differentiability of eigenvalues of positive-definite symmetric matrices

Let $A\in M(n,\mathbb{R})$ be an invertible matrix. Consider the (real) eigenvalues $\lambda_1,\cdots,\lambda_n$, in increasing order, of the positive-definite symmetric matrix $A^t A$. We shall ...
6
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1answer
456 views

General Tarski-Seidenberg Theorem

The Tarski-Seidenberg Theorem states that the polynomial image of a semi-algebraic set is semi-algebraic. A semi-algebraic subset of a Euclidean space $\Bbb{R}^n$ is by definition a finite union of ...
2
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1answer
80 views

When is a set defined by multivariate polynomial inequalities convex?

Consider the set of real numbers given by $$S = \{(a,b,c,d,e,f,g,h) \in [0,1]^8 : 0 \le \frac{e(g-h)}{b(g-f)} \le 1 \text{ and } 0 \le \frac{e(h-f)}{(1-b)(g-f)} \le 1\}$$ Note that this set can also ...
9
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3answers
369 views

Polynomial inequality of sixth degree

There is the following problem. Let $a$, $b$ and $c$ be real numbers such that $\prod\limits_{cyc}(a+b)\neq0$ and $k\geq2$ such that $\sum\limits_{cyc}(a^2+kab)\geq0.$ Prove that: $$\sum_{cyc}\...
8
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0answers
113 views

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 $...
5
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0answers
91 views

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 ...
18
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0answers
270 views

Is every minimal hypersurface in $S^n$ algebraic?

Let $S^n$ be the round n-sphere. Wu-yi Hsiang asked in his paper “Remarks on closed minimal submanifolds in the standard riemannian m-sphere” (1967) the follow question Is every minimal ...
1
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1answer
89 views

A close-form solution for a simple quadratic optimization problem

Is there any closed form solution for the following optimization problem: \begin{align} &\min_{\mathbf{X},\alpha} \mathrm{Tr}[(\mathbf{A}-\mathbf{B}\mathbf{X})(\mathbf{A}-\mathbf{B}\mathbf{X})^{\...
9
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1answer
501 views

Do these surfaces intersect?

For any real numbers $a_{1},a_{2},\cdots a_{6}$ and $b_{1},b_{2},\cdots b_{6}$ with $\sum_{i=1}^{6}a_{i}^{2}=1$ and $\sum_{i=1}^{6}b_{i}^{2}=1$, does the equation $$ x_{1}^{2}x_{2}^{2}x_{3}^{2}x_{4}^{...
3
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0answers
124 views

Real analytic function: zero set of the gradient is a subset of the zero set of the function

I had this question when reading Bierstone and Milman's famous paper "Semianalytic and subanalytic sets". In their proof of the Łojasiewicz gradient inequality (Proposition 6.8 in the paper), they ...
2
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2answers
269 views

Why does a complex linear normalization of a real algebraic surface inherit a real structure?

Could you recommend any references to (some of) the following very basic assertions in algebraic geometry? (It seems unreasonable to reprove them in a research paper.) (1) Let a surface $X$ in $\...
7
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1answer
406 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 $\...
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0answers
136 views

Quantifier elimination with no free variables and real polyhedral inequalities

In this introductory blog post https://cstheory.blogoverflow.com/2011/11/something-you-should-know-about-quantifier-elimination-part-i/ it is mentioned in the very last line that "I do not know if a ...
3
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1answer
305 views

Lower bound for polyhedral real quantifier elimination

All known examples for double exponential lower bounds for real quantifier elimination involves polynomial inequalities with degree $>1$. Is there an example of double exponentiality with ...
2
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2answers
107 views

How to determine the range of values ​of A(i,j) in Covariance matrix A?

Let $A(i,j), i,j=0,1,2$ be the covariance matrix of three random variables. If we know all the entries except $A(2,0)$ and $A(0,2)$, how to determine the range of possible values of $A(2,0)$?
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1answer
151 views

Homogeneous polynomial in 4 variable with non degenerate zero

I've got a very simple question about a homogenous polynomial, for which I cannot see neatly how to proceed (probably due to my limitations in algebraic geometry though). Any help would be greatly ...
1
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1answer
108 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 ...
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2answers
247 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,\...
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0answers
122 views

Example of applying real quantifier elimination algorithm for polynomials

Sorry if any of this is unclear, or doesn't make much sense, I'm still trying to figure it out, a practical example such as this would likely help me understand better than anything else. I have read ...
8
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0answers
134 views

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 ...
3
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1answer
78 views

Intersection of quadratic equations with planted solutions?

Suppose we have two quadratic equations in $\mathbb R[x_1,\dots, x_n]$. What is the expected dimension of their intersection? In general what can we say about intersection of $k$ quadratics? How many ...
2
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1answer
87 views

SDP representation of ideal polynomials for positivstellensatz refutations

If we want to certify the nonexistence of real solutions to a polynomial system of equations, i.e. $$ S = \{ x\in \mathbb{R}^n \ | \ h_i (x) = 0, \ i=1,\dots,t \} = \emptyset, $$ we can produce a ...
5
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1answer
189 views

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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1answer
203 views

Does quantifier elimination help here?

Suppose we have a quantified linear program $$\exists z_1,\dots,z_{poly(n)}\in\mathbb R$$ $$\exists u_1,\dots,u_n\in\mathcal P\cap\mathbb R^m$$ $$\forall v_1,\dots,v_n\in\mathcal P\cap\mathbb R^m$$ $$...
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0answers
54 views

A retract algebraic subset of the plane which does not admit an algebraic retraction

What is an example of an algebraic (=Zariski closed) subset $C$ of $\mathbb{R}^2$ which is a topological retract of $\mathbb{R}^2$, but there is no algebraic retraction $P:\mathbb{R}^2 \to C$? What ...
9
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1answer
230 views

Is it possible for the Witt group of a scheme to have non-trivial odd torsion?

Let $F$ be a field of characteristic not $2$. A well-known theorem of Pfister asserts that the torsion of $W(F)$, the Witt group of $F$, is $2$-primary. Baeza [B, V.6.3] extended this result to Witt ...
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0answers
159 views

Kronecker's theorem in higher dimension

Recall the following classical theorem of Kronecker: if $P(x) \in \mathbb{Z}[x]$ is a monic irreducible polynomial with all roots on the unit circle $S^1$, then $P(x)$ is a cyclotomic polynomial (and ...
5
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0answers
341 views

Are nearby points in an algebraic curve necessarily connected?

I would like a result of the following form: For every algebraic curve $C$ in $\mathbb{R}\mathbf{P}^{n-1}$, there exists an explicit and easy-to-compute $\epsilon=\epsilon(C)>0$ such that ...
6
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1answer
241 views

Is every basis for $\bigwedge^kV$ satisfying a “complementary” property a rescaling of a “standard” basis?

This is a cross-post. Let $V$ be a $4$-dimensional real vector space. Let $\omega_{i_1,i_2}$ be a basis for $\bigwedge^2V$, where each $\omega_{i_1,i_2}$ is decomposable. Suppose that for every $\...
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0answers
105 views

Orbits of unipotent groups

Let $V$ be a real vector space (of finite dimension) and let $G$ be a unipotent Lie subgroup of $\mathrm{GL}(V)$. The orbits of points under the action of $G$ (that is, the sets $Gx = \{T(x) \ : \ T \...
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0answers
38 views

Sum of squared nearest-neighbor distances between points on the sides of a rectangle

For positive real numbers $a,b$, let $R$ denote the $a\times b$ rectangle $[0,a]\times[0,b]$. Let $A_1,\dots,A_4$ be points on the sides of $R$, one point on each side. For each $j=1,\dots,4$, let $...
3
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0answers
59 views

Biggest Cartesian Product Included in a Real Plane Curve

Suppose an irreducible smooth $p \in \mathbb{R}[x_1,x_2]$ is given, and we would like to find finite sets $S_1 , S_2 \subset \mathbb{R}$ such that $p(S_1 \times S_2)=0$ and $|S_1 \times S_2|$ is as ...
6
votes
1answer
214 views

Number of connected components of degree 2 affine algebraic varieties

Suppose an algebraic variety $V$ is given as the solutions to $q$ polynomial equations of degree $\le k$ with real coefficients $$p_1(x_1,\dots,x_m)=0,\dots,p_q(x_1,\dots,x_m)=0$$ for $x\in\mathbb R^m$...
2
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1answer
153 views

About independence spread

$A$, $B_{i}$ are some events. If $A$, $B_{i}$ are independent $\forall i \in \mathbb N$ and $A \cap B_{1}, A \cap B_{2}, ..., A \cap B_{k}, ...$ are independent in aggregate, how to show, that $\...
16
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1answer
494 views

Cohomology of real analytic coherent sheaves

Let $M$ be a real analytic variety (if someone is concerned about distinction between "real analytic spaces" and "real analytic varieties" in real analytic geometry, let's assume that $M$ is both "...
8
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0answers
225 views

Maximum number of connected components of a real affine curve

Harnack's curve theorem tells us that the maximum number of connected components of an algebraic curve of degree $d$ in the real projective plane is $1 + (d-1)(d-2)/2$ (and this bound is sharp). What ...
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0answers
42 views

Affine automorphisms of real affine varieties

Let $V \subset \mathbb{R}^d$ be a real affine variety. I'm hoping I will not butcher existing nomenclature too badly if I say that for the purposes of this question an affine automorphism of $V$ is an ...
10
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2answers
167 views

The set of polytopes with given $f$-vector

Let $f=(f_0,\ldots f_n)$ be a vector in $\Bbb N^{n+1}$. Let $X$ be the set of all (ordered) $f_0$-tuples in $\Bbb R^n$ whose convex hull has $f$ as its $f$-vector. Assume that $X$ is non-empty. Is ...
7
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1answer
117 views

another extremal property of regular polygons

$\newcommand{\R}{\mathbb{R}}\newcommand{\D}[1]{\Delta_{#1}}\newcommand{\set}[1]{\{#1\}}\newcommand{\abs}[1]{\lvert#1\rvert}\newcommand{\E}{\mathbb{1}}$ In 1984 S.D.Berman, a Soviet mathematician, ...
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0answers
122 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 $...
11
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0answers
208 views

Matrices that admit a power that is symmetric

We fix an integer $n\geq 2$. Let $S_n$ be the set of real symmetric matrices in $M_n(\mathbb{R})$. We consider the algebraic sets $Y_k=\{A\in M_n(\mathbb{R});A^k\in S_n\},k\geq 2$ and the sequence $...
18
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2answers
634 views

How can I distinguish a genuine solution of polynomial equations from a numerical near miss?

Cross-posted from MSE, where this question was asked over a year ago with no answers. Suppose I have a large system of polynomial equations in a large number of real-valued variables. \begin{align} ...
19
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1answer
593 views

Counting real zeros of a polynomial

I recently came across a criteria to count the number of real zeros of a polynomial $P(x)$ with real coefficients. Unfortunately I cannot find the reference! The criteria is the following: Form the ...
5
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1answer
93 views

(Euclidean) open orbit in an irreducible real algebraic set

Let $\tau:GL(n,\mathbb{R}) \rightarrow GL(V)$ be a rational representation of the general linear group of degree $n$ on a finite-dimensional real vector space $V$. Let $C$ be an irreducible real ...
3
votes
1answer
121 views

Atoric equation

I'm looking for a general equation/function z = f(x, y, radius1, radius2, p1, p2) for an atoric surface. p1 and p2 could be either eccentricity or conic constant values. Can anyone help me with that? ...
2
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1answer
170 views

Symmetric orthogonal matrices with constant diagonal entries

$\newcommand{\al}{\alpha} \newcommand{\be}{\beta} \newcommand{\de}{\delta} \newcommand{\De}{\Delta} \newcommand{\ep}{\varepsilon} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\la}{\...
7
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1answer
396 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 ...
3
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1answer
167 views

Solutions to a system of homogeneous equations (inequalities)

Let $f_1,\ldots,f_r \in \mathbb{R}[x_1,\ldots,x_n]$ be $r$ homogeneous polynomials of the same odd degree $d$, where $d \in \{3,5,7,\ldots\}$. For which values of $r,n,d$ there exists a real ...

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