# 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).

214 questions
Filter by
Sorted by
Tagged with
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 ...
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 ...
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 ...
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 ...
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 ...
369 views

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 ...
269 views

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 ...
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 ...
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 $A(2,0)$?
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 ...
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 ...
247 views

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 "...
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 ...
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 ...
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 ...
117 views

### another extremal property of regular polygons

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

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} ...
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 ...
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 ...