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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185 views

Rationality of a certain real algebraic variety

Let $A_n$ denote the vector space of $n\times n$ antisymmetric matrices over ${\mathbb{Q}}$, where $n$ is even. Let $A_n^*\subset A_n$ denote the affine ${\mathbb{Q}}$-subvariety of invertible ...
11
votes
4answers
317 views

Tameness in $\mathbb{R}^{n^2}$ of the subset consisting of matrices of positive determinant

The Lie group $GL(n)$ being a manifold is locally path-connected. Consider its connected component of the identity $C\subseteq\mathbb{R}^{n^2}$. What is a good way of showing that $C$ is a tame ...
1
vote
2answers
107 views

Smooth, irreducible surface with real part containing two projective planes

Let $X$ be a smooth and irreducible projective variety over $\mathbb{R}$ of dimension two. I am looking for an instance of such a variety where two distinct connected components of $X(\mathbb{R})$ are ...
3
votes
0answers
67 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 ...
2
votes
0answers
101 views

Log Canonical Threshold Equals 1

I'm an analyst interested in what we would call the "critical integrability index" which, for a given (real-valued) function $f$ is the supremum of $\sigma$ such that $|f|^{-\sigma}$ is integrable on ...
3
votes
1answer
72 views

ideal generated from a truncated “real radical-like” set are still real radical?

Suppose $\mathbb{R}[X]$ is the normal multivariate polynomial ring where $X = x_1...x_n$. $\mathbb{R}[X]_t$ is the truncated ring such that $\mathbb{R}[X]_t =\left\{f: f \in \mathbb{R}[X], deg(f) \leq ...
2
votes
0answers
57 views

Upper bound on the number of convex connected components of the complement of the zero set of a polynomial

The classic Milnor-Thom upper bound on the sum of the Betti numbers of real algebraic sets (for a nice exposition and references, see e.g. N.R. Wallach, On a Theorem of Milnor and Thom, in S. Gindikin ...
4
votes
0answers
156 views

Does the cohomology comparison part of GAGA hold over the reals?

If $k=\mathbb{C}$, $X$ is a projective (or even proper) scheme over $k$ and $F$ a coherent sheaf on $X$, then there is an isomorphism $$ H^p(X,F)\rightarrow H^p(X^{an}, F^{an}) $$ (and there is also ...
1
vote
0answers
75 views

Irreducible real curves on ${\mathbb C}P^1$ invariant under the finite group action

Let $G$ be a finite subgroup of a Möbius group with a standart action on a real algebraic variety ${\mathbb C}{\mathbf P^1}.$ How one can describe $G$-invariant irreducible real algebraic curves? ...
9
votes
3answers
290 views

Decay of real continuous algebraic functions at infinity

Let $f$ be a real valued continuous algebraic function on $\mathbb R^n$. Suppose the zero set of $f$ is bounded, i.e., if $|x|$ is large enough, $f(x)\neq 0$. Is there any estimate of the sort ...
4
votes
0answers
157 views

Plane real curves such that their intersections with lines are hyperbolic

Let $R$ be an (irreducible) plane real algebraic curve (without isolated points). Consider Zarissky closure of $R$ in ${\mathbb C}P^1$ (as real variety). Suppose that $\lambda\in R ...
7
votes
1answer
296 views

Geometry of Hermitian rank $\leq r$ matrices

Let $M_n^{sa}$ be the space of $n\times n$ complex Hermitian matrices, let $r < n$, and let $E$ and $F$ be (real) linear subspaces of $M_n$ with ${\rm codim}(E) < r^2$ and ${\rm codim}(F) = 1$. ...
2
votes
1answer
73 views

Could a real curve symmetric across the line be defined only by polynomial that is not reflection-invariant?

Let $R$ -- be an irreducible plane real algebraic curve (without isolated points). Suppose that $(x,y)\in R\Leftrightarrow (x,-y)\in R.$ Question: could one find a polynomial $f(x,y)$ with zero set ...
4
votes
0answers
91 views

Where do I read about semi-algebraic/analytic sets?

What's a good introduction to semi-algebraic/semi-analytic sets? I'm coming from algebraic geometry, so ideally I'd prefer material that has a similar point of view and highlights analogies. I've ...
3
votes
0answers
94 views

What subdomains of $\mathbb{R}^2$ are diffeomorphic to $\mathbb{R}^2_+$ via rational functions?

For what subdomains $D \subset \mathbb{R}^2$ does there exist a rational diffeomorphism $h:D \to \mathbb{R}^2_+$, such that its inverse is also a rational function? (By "rational function" I mean a ...
4
votes
1answer
108 views

Invariant regular cones in Lie group representation

I am following Analysis and Geometry on Complex Homogeneous Spaces by Faraut et al. I'll set up all of what I need and then ask my questions. Let $G$ be a connected semi-simple non-compact real Lie ...
2
votes
1answer
106 views

When is the image of the adjoint representation of a real algebraic group Zariski closed?

Let $\operatorname{Ad}:\operatorname{SL}_n(\mathbb{R}) \to \operatorname{GL}(\mathfrak{sl}_n(\mathbb{R}))$ be the adjoint representation (i.e. $\operatorname{Ad}(g)X=gXg^{-1}$) of $SL_n(\mathbb{R})$. ...
2
votes
0answers
84 views

What is known about topological equivalence of polynomial dynamical systems on two different domains in R^n?

The question is mainly about $\it flows$, not maps (i.e., continuous time, not discrete time). Is it known if the study of polynomial dynamical systems on $\mathbb R^n$ can be reduced to the study ...
10
votes
1answer
234 views

Euler Characteristic of Real Algebraic Surfaces

Given a (compact if needed) real smooth surface $V(f)$ defined by $f\in \mathbb{R}[X,Y,Z]$, in particular it is oriented. Is there a formula which gives the Euler character of $V(f)$ ? Thanks.
8
votes
2answers
905 views

Incidence geometry and matrices

Supposing I have a $0/1$ or $\pm1$ matrix $A$ of size $m\times n$, is there a minimum $d$ (that works for every $m\times n$ $A$) such that there exists $m$ lines $r_1,\dots,r_m$, $n$ lines ...
2
votes
0answers
51 views

Algebraic independence in normed spaces

A set of $n$ points in $\mathbb{R}^2$ is algebraically independent over $\mathbb{Q}$ if there is no polynomial dependency among the $2n$ coordinates. A result (Lemma 3.3) from "Globally linked pairs ...
3
votes
0answers
181 views

Number of real singular points

Let $f\in\mathbb{R}[X,Y]$ be a real polynomial in two variables. Are there bounds on the number of singular points of $f$ which take into account that $f$ might have high degree but be rather sparse? ...
10
votes
2answers
474 views

The space of polynomials with all real roots

The question stems from an attempt to answer a question of David Speyer. Let $R \subseteq \mathbb{R}^n$ be the space of coefficients of all polynomials of degree $n$ whose all roots are real, i.e. $R ...
1
vote
0answers
71 views

Proximal point avoids a subvariety

I have a seemingly easy problem about the proximal point in a variety of a general point in the ambient space, which I don't have a proof: Let $X \subset \mathbb{C}^N$ be the affine cone over some ...
2
votes
1answer
121 views

Real algebraic surface

Let $f(u,v,z)\in \mathbb{Q}[u,v,z]$ be a polynomial in three variables such that $X_{\mathbb{R}}\subset \mathbb{R}^{3} $ (the associated surface of real solution) is smooth. Suppose that the set of ...
2
votes
1answer
80 views

Polynomial degree comparison of Nullstellensatz and Positivstellensatz over real algebraic sets

Suppose we have a (finite) system of polynomials $P = \{ p_i \} \subseteq \mathbb{R}[x_1, \ldots, x_n]$. Then it is well known by the Nullstellensatz that either $P$ has a simultaneous zero over ...
1
vote
1answer
67 views

About convex combinations of real-stable multivariable complex polynomials

Say $f: \mathbb{C}^{n+1} \rightarrow \mathbb{C}$ is a real stable multivariable polynomial on the variables $(z,w_1,w_2,...,w_n)$. (a "real-stable" polynomial is one which has no zeroes in the open ...
6
votes
1answer
483 views

Bound on the sum of arguments

Problem: Show that for all real $s,t,u$ and all complex $z$ with $|z|<1$ one has $$(*)\qquad \arg\frac{1-zf(s-u)}{1-zf(s+u)} +\arg\frac{1-zf(t+u)}{1-zf(t-u)}<\pi, $$ where $f$ is the ...
2
votes
2answers
141 views

About preserving real-rootedness of multivariable polynomials

Say $f_i(z_1,z_2,..,z_m)$ are polynomials real rooted in the $z$s for a bunch of polynomials indexed by $i$. When can one say that $\sum_{i} p_i f_i(z_1,z_2,..,z_m)$ is also real rooted? If ...
2
votes
0answers
122 views

Real-rooted polynomials and higher rank matrices

For $A$ and $B$ being matrices of the same dimension and $B$ being rank $1$, one knows that $det(A+tB)$ is a linear polynomial in $t \in \mathbb{R}$. Hence by Taylor series it follows that $det(A + ...
1
vote
1answer
64 views

image under a morphism of a variety defined over R

If you have say an affine variety defined over $\mathbb{R}$, then its image under a morphism (also defined over $\mathbb{R}$) is a constructible set. But presumably there would be no good reason in ...
0
votes
0answers
141 views

Homogeneous polynomials of degree 3 in two variables

Let $V$ be the four-dimensional real vector space consisting of all homogeneous polynomials of degree 3 in two variables with coefficients in $\mathbb{R}$. Let $U$ be the set of all elements of $V$ ...
2
votes
1answer
159 views

Determine whether a system of polynomials with real coefficients has a real solution?

Today my students asked me the following problem: Define polynomials $P_j$ with real coefficients $$P_{j}=\sum_{i_{1},\cdots,i_{s}}a^{(j)}_{i_{1}\cdots i_{s}}X^{i_{1}}_{1}\cdots X^{i_{s}}_{s}, ...
2
votes
0answers
68 views

conic structure at infinity for non-closed unbounded semi-algebraic sets

Let $X\subseteq\mathbb{R}^k$ be a non-closed, unbounded semi-algebraic subset. Then it seems to me that Proposition 5.49 on p. 189 of the book Algorithms in Real Algebraic Geometry still holds true ...
7
votes
0answers
136 views

Nearest Point to a real algebraic set

Suppose I have a compact bounded real algebraic (eventually: or analytic or semialgebraic or semianalytic set) $V$ in $\mathbb R^3$ and a point $x\in\mathbb R^3$ not in $V$. How much do we know about ...
-1
votes
1answer
156 views

Smooth algebraic functions

Suppose $f(x)=\sum_{|\alpha|=0}^{\infty}a_{\alpha} x^{\alpha}$ for all $x\in\mathbb{R}^n$. Moreover we know a priori that $f$ is an algebraic function. Is $f$ necessarily a polynomial?If not what are ...
0
votes
0answers
100 views

Dimensions of two spaces

Let $R=\Bbb K[x_1,\dots,x_n]/I$ and $Z=\mathsf Z(I)$ where $\Bbb K$ is any field with $char(\Bbb K)\neq 2$. Is there a way to describe space of $f_1,f_2\in R$ that satisfies $$f_1+f_2\neq 0,\mbox{ ...
2
votes
1answer
199 views

How many circular cylinders through 5 general points?

How many right circular cylinders can pass through 5 general points in ℝ3 ? Edit: 0,2,4, or 6.
2
votes
3answers
220 views

Characterizing orthants with polynomials

Let $x=(x_1,\ldots,x_n)\in\mathbb{R}^n$. Can one find a polynomial $p$ (of arbitrary degree) in the coordinates of $x$ such that $p(x)\geq 0$ if and only if $x$ is an element of the positive orthant ...
11
votes
0answers
318 views

Does every commutative $*$-algebra of operators on a prehilbert space have a character?

My question can be equivalently stated as follows: Let $A$ be a complex commutative algebra with involution, and assume there exists a non-zero homomorphism $\pi:A\to L(V)$ to the algebra of ...
4
votes
1answer
280 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 ...
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vote
0answers
78 views

vector space of ternary forms with real rooted property

Let $V \subseteq \mathbb{R}[x,y]_d$ be a two dimensional linear subspace of the vector space of bivariate forms of degree $d$. For each degree $d$ we can find such subspaces with the property that ...
6
votes
1answer
274 views

Non-negativity of invariant polynomials

Certifying the non-negativity of a symmetric polynomial is much easier than certifying the non-negativity of an arbitrary polynomial function: for instance, in (1) is proved that the complexity of ...
3
votes
0answers
60 views

Real centrally-symmetric plane algebraic curves

I am looking for a reference regarding the topology of real centrally-symmetric plane algebraic curves. By this I mean the curves defined by $$ P(x,y)=0, $$ where $P$ is a degree $m$ polynomial, ...
2
votes
0answers
118 views

Behavior of elementary symmetric polynomials near zero sets

It is straightforward to show (see Characterizing intersection of zero sets of elementary symmetric polynomials on R^n) that the set of points $\Lambda_{k}$ in $x \in \mathbb{R}^{n}$ with ...
5
votes
1answer
252 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 ...
3
votes
1answer
155 views

Handelman's positivstellensatz for symmetric matrix-valued polynomials

For certain classes of sets $S \subseteq \mathbb{R}^n$, there exist algebraic characterizations of real valued polynomials $p: \mathbb{R}^n \rightarrow \mathbb{R}$ that are positive on $S$. Several ...
3
votes
1answer
123 views

Tropical polynomial Positivstellensatz

In real algebraic geometry, Stengle's Positivstellensatz can be used to characterize polynomials that are positive on a semialgebraic set. Say that a tropical semialgebraic set is a subset of ...
7
votes
3answers
327 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 ...
5
votes
1answer
266 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 ...