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4
votes
2answers
494 views

intersections of real algebraic sets (a bezout-type question)

This question arose when I was trying to understand the Guth-Katz paper on Erdos distance problem, namely, related to Proof of Lemma 2.9 in http://arxiv.org/abs/1011.4105. The proof of that lemma is ...
3
votes
1answer
459 views

Quick algorithm for finding real solutions for a system polynomial equations

Would you please recommend a computer program that could give quick answer Yes/No for the question: does there exist a real solution of a given system of polynomial equations with integer ...
8
votes
2answers
415 views

The topology of open semi-algebraic sets (appl.: totally positive matrices)

Let $P_1,\ldots,P_r$ be polynomials over ${\mathbb R}^N$. I am interested in the homotopy type of the semi-algebraic set defined by $$ P_j(x_1,\ldots,x_N)>0,\qquad j=1,\ldots,r . $$ Is there a ...
2
votes
0answers
240 views

Integrating over the real points of an algebraic variety

This question is motivated by my earlier question. Suppose I have some algebraic variety $X/\mathbb R$ of dimension $n$, and suppose that $X(\mathbb R)$ is compact. Now for any element of ...
6
votes
2answers
577 views

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?
7
votes
1answer
562 views

(Real) algebraic geometry for (real) trigonometric polynomials?

Has somebody developed a comprehensive theory of the algebraic structure of trigonometric polynomials in several variables? If yes, where? Background: By a (real) trigonometric polynomial in ...
2
votes
0answers
268 views

Maximal compact subgroup of a real semisimple Lie group of “quasi-adjoint” type.

Let $\mathfrak{g}$ be a semisimple complex Lie algebra, and $G$ the adjoint group of $\mathfrak{g}$. Let $\sigma:\mathfrak{g}\rightarrow \mathfrak{g}$ be a complex conjugation and ...
3
votes
1answer
237 views

Hypersurfaces without real points

Let $n, d$ be positive integers. I am interested in the open subset $\mathcal U_{n,d} \subset \mathbb P H^0 ( \mathbb P^n_{\mathbb R}, \mathcal O_{\mathbb P^n_{\mathbb R}}(d))$ corresponding to ...
12
votes
5answers
1k views

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 ...
2
votes
3answers
715 views

Isolated zero set of real polynomial in two variables

The polynomial $x^2+y^2$ has an isolated zero at the origin. And so do powers $(x^2+y^2)^n$ of this polynomial. I'm wondering if this is a special property of these real polynomials. Here's the ...
6
votes
2answers
598 views

Conditions on coefficients of a homogeneous polynomial of the third degree in three variables over R which allow it to be positive on a positive octant

I am a student of Saint Petersburg State Polytechnical University, chair of Theoretical Mechanics. While looking into stability of ideal crystal lattices (2D and 3D) by means of molecular dynamics I ...
10
votes
3answers
1k views

Counting roots: multidimensional Sturm's theorem

Sturm's theorem gives a way to compute the number of roots of a one-variable polynomial in an interval [a,b]. Is there a generalization to boxes in higher dimensions? Namely, let $P_1,\dotsc,P_n\in ...
3
votes
1answer
417 views

Function Fields of Real Varieties

Let $V$ be a geometrically irreducible and reduced scheme defined over the real numbers, and let $K = K(V)$ be its function field. If $V$ does not have any real points, is it true that $K$ is not ...
11
votes
5answers
2k views

Polynomial positive on an interval

If $p$ is a polynomial with real coefficients and p(x)>0 on [0,1], then $p(x)=\sum c_{i,j} x^i(1-x)^j$ with $c_{i,j}$ positive. I know this is true but but I need a proof/reference. Thanks!
10
votes
3answers
446 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 ${\mathbb{R}}^n$?
6
votes
2answers
663 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 ...
3
votes
4answers
1k views

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 ...
19
votes
3answers
2k views

Real algebraic geometry vs. algebraic geometry

This question is predicated on my understanding that real algebraic geometry (henceforth RAG) is the version of algebraic geometry (AG) one gets when replacing (esp. algebraically closed) fields with ...
2
votes
2answers
376 views

Real spectrum of ring of continuous semialgebraic functions

Let R be a real closed field, and let U be a semialgebraic subset of $R^n$. Let $S^0(U)$ be the ring of continuous R-valued semialgebraic functions. Also let $\tilde{U}$ be the subset of Spec$_r ...
24
votes
6answers
2k views

Are all polynomial inequalities deducible from the trivial inequality?

I remember learning some years ago of a theorem to the effect that if a polynomial $p(x_1, ... x_n)$ with real coefficients is non-negative on $\mathbb{R}^n$, then it is a sum of squares of ...
13
votes
2answers
1k views

The Sylvester Gallai Theorem and Sections of Varieties with “Simple Topology”.

The Sylvester-Gallai theorem asserts that for every collection of points in the plane, not all on a line, there is a line containing exactly two of the points. One high dimensional extension ...