# Tagged Questions

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### Bounding volume of cell in complement of zero set

I am given an integer polynomial $f \in \mathbb{Z}[X_1, \ldots, X_n]$ of bounded degree and bounded coefficient size. The polynomial's zero set partitions $\mathbb{R}^n$ into cells. What I am looking ...
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### real algebraic geometry software?

Does anyone have suggestions/experience for any software packages to study real algebraic varieties (for example, counting connected components of hypersurfaces, figuring out the topological type of ...
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### Real varieties with enough algebraic loops

Let $(X,\sigma)$ be a complex variety with complex conjugation (equivalently, an algebraic variety over $\mathbb R$). We use the notations $X(\mathbb R):=X^\sigma$ for the set of fixed points of $X$ ...
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### moduli in real/semi algebraic geometry

Is there a moduli space in semialgebraic geometry analogous to the Hilbert scheme in algebraic geometry? The sort of thing I am imagining is an object in a category of semischemes: Ordinary schemes ...
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### What is the longest algebraic curve?

Consider a convex body $\Omega\subset \mathbb{R}^2$. Let $L(d)$ be the maximum over all curves $C$ of degree $d$ of the length of $C\cap\Omega$. Is $L(d)\leq d P(E)/2$, where $P(E)$ is the ...
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### Simple criterion to verify that the real zeros are an irreducible algebraic set

Given an irreducible polynomial $p$, its set of real zeros might be a reducible algebraic set. For example: $p=(x^2-1)^2 +(y^2-1)^2$. Is there a simple sufficient condition on $p$ so that its real ...
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### Homogenous polynomial inequalities to prove positivity?

If I have $n$ variables $x_1, \cdots, x_n$, and a set $S$ of inequalities of the form $p(x_1, \cdots, x_n) > 0$ where $p$ is a homogenous real polynomial, is it true that I need $|S| \geq n$ to ...
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### Are there general position results in singular algebraic sets?

Let $X$ be a real algebraic set, and let $Y \subset X$ be its singular set. In this question I'll focus on the analytic topology, so we can just imagine that $X$ is the zero set, in $\mathbb{R}^n$, ...
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### Volumes of families of semialgebraic sets

Let $f_0(T),f_1(T) \in \mathbb R [T]$ be polynomials. Let $F(T,X):=X^2+f_1(T)X+f_0(T)$. Then the set $M(t):=\{x \in \mathbb R \mid F(t,x) \le 0\}$ is bounded. Its volume $V(t)$ is ...
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### How to tell if a second-order curve goes below the $x$ axis?

Suppose we have a second-order curve in general form: (1) $a_{11}x^{2}+2a_{12}xy+a_{22}y^{2}+2a_{13}x+2a_{23}y+a_{33}=0$. I'd like to know if there is a simple condition that ensures that the curve ...
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### Real Pfaffian representations of real cubic surfaces

Consider the following classical construction (which is called Pfaffian representation as Sasha indicates): Let $V^4\subset \Lambda^2 \mathbb R^6$ be a four-dimensional subspace of the space of ...
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### orderings of the field R((x, y))

I don't know much about the theory of ordered fields. But I know that, for the real fields $\mathbb{R}(y)$, $\mathbb{R}((x))(y)$, and $\mathbb{R}((x))((y))$, we can explicitly determine all the ...
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### Real vs complex surfaces

Hi! My background is in (complex) algebraic geometry. For the first time I'm considering varieties defined over $\mathbb{R}$ and I have some very basic questions about these. In particular I'm trying ...
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### Hausdorff distance on algebraic curves

Introduction Let $K$ be a nice closed domain in $\mathbb{R}^2$, for example the closed unit ball. Recall that the Hausdorff distance on the family $C(K)$ of nonempty compact subsets of $K$ is defined ...
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Let $R$ be a real closed field and $f: R \to R$ a map. Then let $\textrm{F}(f)$ be the set of semialgebraic subsets of $R^2$, which contain $(t,f(t))$ for all $0< t< \epsilon$ for some $\epsilon ... 1answer 582 views ### Is the pairing between contours and functions perfect (modulo the kernel given by Stokes' theorem)? Let$s: \mathbb C^n \to \mathbb C$be a homogeneous degree-$d$polynomial which is nonsingular (in the sense that the hypersurface it defines in$\mathbb{CP}^{n-1}$is smooth; equivalently the ... 1answer 312 views ### Degree of a real algebraic variety and regular morphisms I'm reading Fulton's "Intersection theory", which i need for some applied needs. And i have two questions on general definition of degree used in Fulton. 1)Let us we have a real algebraic variety ... 1answer 329 views ### Number of connected components of complement to a reducible real algebraic hypersurface.[EDITED] Let$X_1,\ldots X_k$be irreducible(may be singular) affine real algebraic hypersurfaces in$R^n$with$x_1,\ldots, x_k$connected components, respectively. Let$G_1,\ldots, G_l$be their ... 1answer 455 views ### Quick algorithm for finding real solutions for a system polynomial equations Would you please recommend a computer program that could give quick answer Yes/No for the question: does there exist a real solution of a given system of polynomial equations with integer ... 2answers 410 views ### The topology of open semi-algebraic sets (appl.: totally positive matrices) Let$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 ... 0answers 236 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 ... 2answers 543 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? 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 ... 3answers 708 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 ... 2answers 585 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 ... 1answer 413 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 ... 2answers 642 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 ... 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 ... 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 ... 2answers 372 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 ...
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 ...