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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Does anyone have Delzell's Thesis on Bad Points of Forms?

Since a number of papers (e.g. this one) treating denominators in Hilbert's 17th problem point to E.G. Strauss's unpublished letter to G. Kriesel or to Chapter 5 of Delzell's Thesis, which contains an ...
8
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1answer
368 views

Polynomial approximations of curves

This is the 3D version of this question. The responses to that question contained a lot of complaints about fuzzy definition of the problem, so I made this new question very narrow and explicit. For ...
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0answers
69 views

Question about radical ideal

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

Sums of hermitian squares in free abelian group algebras and real positive semidefinite matrices

A little context for the following question, first. As Noah Stein notes in a comment below, the present question is closely related to the free semialgebraic geometry studied by Helton and his ...
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0answers
128 views

local analyticity of volumes of slices of semi-algebraic sets

I would like a reference and/or a simple proof using well-known results of the following, which I think is true. (If it's false, I'd like to know that as well of course -- and ideally a way to modify ...
9
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205 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
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4answers
325 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 ...
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2answers
118 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
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71 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
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0answers
105 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
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1answer
90 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 ...
5
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1answer
145 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
157 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 ...
2
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0answers
78 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
293 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 $|f(x)|\...
5
votes
0answers
161 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 \Rightarrow\...
7
votes
1answer
299 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
96 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
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0answers
96 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
110 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
111 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
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0answers
87 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
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1answer
238 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
909 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 $s_1,\dots,...
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
483 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 ...
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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
82 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
69 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
484 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
146 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
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0answers
127 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 + tB)...
1
vote
1answer
66 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
144 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
163 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
70 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 ...
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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
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0answers
101 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
200 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 $\...
12
votes
0answers
322 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
283 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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0answers
79 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
281 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 ...