Questions tagged [real-algebraic-geometry]

Real algebraic geometry is the study of algebraic geometry over the real numbers, or more generally formally real (esp. real closed) fields. Problems in this tag may require a mix of methods from algebraic geometry and techniques from o-minimal (esp. semialgebraic) geometry.

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41 views

Irreducible components of real algebraic sets

Let $\mathbb{R}[x_1, \ldots , x_n]$ denote the commutative ring of all polynomials in $n$ variables $x_1, \ldots, x_n$ with coefficients in $\mathbb{R}$. Given a set with $k$ polynomials $\{f_1, . . . ...
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Criteria for independence of nonlinear system of differential equations

Suppose we have a system of second-order nonlinear differential equations. By this I mean that all the equations are of order less or equal to two, so I allow for situations when some equations are of ...
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1answer
38 views

Recommendations for Real Algebraic Geometry and Optimization

I just started looking into positive polynomials on compact semi-algebraic sets and it requires a mixture of optimization, functional analysis and real algebraic geometry. I would like to know if you ...
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1answer
52 views

Real algebraic set containing a sphere

Consider an irreducible polynomial $P \in \mathbb{R}[x_1,\ldots ,x_n]$ and define $$V := P^{-1}(0) = \left\{ (x_1, \ldots ,x_n) \in \mathbb{R}^n \mid P(x_1,\ldots ,x_n)=0 \right\}$$ It's well known ...
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1answer
48 views

Upper bound for amount of intervals in intersection of interval sets

I have two sets of numbers which are unions of disjoint intervals, and I have to find an upper bound for how many of such intervals can there be in the intersection of the two sets. Here's a diagram ...
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Does a sequence of $d$-SOS polynomials converge to a polynomial that is also $d$-SOS?

Let $\mathbb{R}[X]_{\leq 2d}$ denote the real vector space of polynomials of degree at most $2d$ in the coordinate ring $\mathbb{R}[X]$ of variety $X$. Definition: A polynomial $f$ is $d$-SOS if ...
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1answer
72 views

The quotient scheme $X/\Gamma$ when $X$ is separated and every orbit is contained in an affine.

I am trying to solve Problem II.4.7(a) of Hartshorne: The only candidate I can think of for $X_0$ would be the quotient scheme $X/\sigma$. If it exists, it must be unique by the usual argument. First ...
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When does the (real) zero locus $F(x,y,t) = 0$ “look the same” for all values of the parameter $t$?

I have a problem where I would like to know when the zero locus of a deformed polynomial "looks like" (by "looks like" I mean homeomorphic but I am not sure if that is the correct ...
6
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1answer
97 views

Reference for a real algebraic geometry problem

Disclaimer: I am not a mathematician by training. I encountered the following problem in my research. Assume that I have $N$ real variables $x_1, x_2, \dots, x_N$. I am given $N$ homogeneous ...
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25 views

Is there any transformation that connects functions whose extremum are same?

I wonder if there is a transformation between functions whose extremum are same, finding extremum of general function becomes easy a bit, but is it known already? For example, $f(x) = x^2$ has its ...
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29 views

Any good usable software or software package for decomposition with constraint?

Does anyone know any good and usable software package, preferably in Windows, that can effectively find SOS (sum of squares) polynomials $s_{0}\left(x\right)$ and $s_{1}\left(x\right)$ for any real ...
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1answer
41 views

Complex varieties as real affine varieties: how to recover complex structure?

If $A$ is a finitely generated $\mathbb C$-algebra without nilpotents, then $A = \mathbb C[V]$ is the $\mathbb C$-algebra of polynomial functions on $V := \mathrm{maxSpec}(A)$ (this is precisely the ...
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Separating Points and Tangent Vectors (real curves)

In [Hartshorne, Proposition 7.3.] as well as in [Görtz & Wedhorn, Rem. 13.55] and [Vakil Notes, around 19.2] the following is said: If $X$ is a curve over (let's say) $\mathbb{C}$ (algebraically ...
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22 views

Formula/Method analyse xy graph points

I need to program in C to analyse a xy graph points with respect to s rectangular box drawn inside the xy graph. Imagine an empty xy graph sheet with a rectangular box drawn somewhere in it. Now we ...
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47 views

How to effectively find the first polynomial of the regular chains?

Could anyone help me with how to, perhaps by using software or programming, effectively find the first polynomial of the regular chains or triangulation factoring from a Grobner basis that has many ...
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1answer
74 views

Real Nullstellensatz

I am reading the proof of the Real Nullstellensatz (from the Real Algebraic Geometry book, by Bochnak, Coste and Roy). I add a picture of this proof And there is a notation I had not seen before: the ...
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1answer
110 views

Is the exponential function semi-algebraic?

Recall the following definitions: We say a set $E\subseteq\mathbb{R}^n$ is semi-algebraic if there exist real polynomials $g_{ij},h_{ij}:\mathbb{R}^n\rightarrow\mathbb{R}$ such that $$E=\bigcup_{j=1}...
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1answer
53 views

Is it possible to send an element in an ordered real algebra both to a positive unit and to a negative unit?

Let ${(A, P)}$ be a preordered $\mathbb{R}$-algebra in the sense that $A$ is a $\mathbb{R}$-algebra and ${P \subseteq A}$ is a subset closed under addition, multiplication, containing the nonnegative ...
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57 views

Real algebraic sets defined by polynomials with coefficients in $\mathbb{Q}$

I'am looking for a proof or a counterexample of the following conjecture. Let $V\subset\mathbb{R}^n$ be a smooth algebraic set of dimension $m$. The following properties are equivalent: V is given ...
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About embeddings of real grassmannians

I am dealing with the embedding of real grassmannians $G(n,k)$ on $\mathbb{R}^{n^2}$ via the map associating to each vector space the projection matrix on it in the canonical base of $\mathbb{R}^n$. ...
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2answers
123 views

When do two conic sections intersect?

Given two conics $$ax^2+bxy+cy^2+dx+ey+f=0$$ and $$Ax^2+Bxy+Cy^2+Dx+Ey+F=0$$ I want to find the conditions that they have a common intersection point. Unfortunately I think the answer may involve ...
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2answers
52 views

Show that $\mathbb{Q}(\sqrt{5})$ is real closed

Problem: Show that $\mathbb{Q}(\sqrt{5})$ is real closed. There is a useful criterion to solve this problem that is Artin-Schreier criterion. How to apply this to this problem?
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1answer
37 views

Show that extension $\mathbb{R}(x) \supseteq \mathbb{R}$ is a real extension (ordered extension)

Problem: Denote $\mathbb{R}(x)$ be the quotient field of polynomial ring in single variable $\mathbb{R}[x]$. Show that extension $\mathbb{R}(x) \supseteq \mathbb{R}$ is a real extension (ordered ...
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1answer
54 views

Show that the real field $\mathbb{R}$ has a unique ordering and indicates that ordering

Problem: Show that the real field $\mathbb{R}$ has a unique ordering and indicates that ordering. My question: We knew that $\le$ is an ordering on $\mathbb{R}$. Do we need to prove that $\le$ is an ...
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1answer
54 views

When do two real polynomials define the same real hypersurface?

Let $V$ be a real affine algebraic subset of a finite-dimensional real affine $n$-space defined by the vanishing of finitely many homogeneous polynomials in $n$ real variables $x_1,\ldots,x_n$. Let $f$...
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64 views

Finitely generated ideals of the ring of smooth functions on a smooth manifold

My question refers to an exercise in "Topology of Real Algebraic Sets" by Akbulut and King. This exercise is about one of the possible definition of blow-up in a differential framework, more precisely:...
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30 views

How to extend an embedding between real closed fields to more large embedding?

Let $\mathbb{H} \subset \mathbb{F}$ be real closed fields and $\mathbb{G}$ be an $\eta_1$ real closed field (i.e. $\mathbb{G}$ has no countable gap ). Fix any countable $Y\subset \mathbb{F}$. Suppose ...
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72 views

When is a plane algebraic curve closed?

When is a plane algebraic curve closed? Is there a metric? For quadratic http://mathworld.wolfram.com/QuadraticCurveDiscriminant.html we know the discriminant has to be below 0. Is there such a ...
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1answer
78 views

Is there a polynomial which detects when the two smallest roots of a given real polynomial are equal?

The discriminant of a polynomial over a field is a "universal"* polynomial function of its coefficients, which is zero if and only if the polynomial has a multiple root in some field extension. Now, ...
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Homology in real closed fields

I'm studing on "Real Algebraic Geometry" by Bochnak-Coste-Roy and in Chapter 11 I found a very interesting characterization of the Euler-Poincaré characteristic of real algebraic sets over a general ...
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64 views

Show that every nonnegative polynomial $P \in \mathbb R [x]$ can be written as a sum of squares of real polynomials

Define a real polynomial $P \in \mathbb R[x]$ to be nonnegative if $P(x) \geq 0$ for all $x \in \mathbb R$. Show that every nonnegative polynomial $P \in \mathbb R [x]$ can be written as a sum of ...
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71 views

General hyperplane sections of real varieties

Definition: A real algebraic variety in $\mathbb{R}^n$ is the set of common real zeroes of some $f_1, \dots , f_s \in \mathbb{R}[x_1, \dots , x_n]$. By the dimension of a real algebraic variety $V$ ...
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50 views

Proving an inequality based on the discriminant of a cubic polynomial

I have a cubic univariate polynomial parameterized by rational numbers $s,t$ $$T^3+Ts + t \in \mathbb Q[T]$$ such that $t$ is in the open interval $]0,1[$ and $s$ satisfies $4s^3+27t^2 > 0$ (i.e. ...
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0answers
37 views

Extension of smooth functions on a locally closed subset of $\mathbb R^n$ to the whole space

Let $V$ be a locally closed subset of $\mathbb{R}^n$, that means the intersection of a Zariski closed subset of $\mathbb{R}^n$ with a Zariski open subset of $\mathbb{R}^n$. My question is how one can ...
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1answer
108 views

On solution to the equation $x_{1}^{2}x_{2}^{2}x_{3}^{2}x_{4}^{2}x_{5}^{2}\left(\sum_{i=1}^{6}a_{i}x_{i}\right)^{2}=1$

For any $a_{1}, a_{2}, \dots, a_{6} \in \mathbb{R}$ with $$\sum_{i=1}^{6}a_{i}^{2}=1$$ is it true that there always exist $x_{1}, x_{2}, \dots, x_{6} \in \mathbb{R}$ with $\displaystyle\sum_{i=1}^{...
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1answer
51 views

Quadratic forms which represent the same element

Let $a,b \in F^\times$. Show that for quadratic forms holds: $$D(\lt 1,a \gt) \cap D(\lt 1,b \gt) \subseteq D(\lt 1,-ab \gt).$$ Here these sets represent the set of elements in $F^\times$ which are ...
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1answer
24 views

Existence proof for the minimal $n \in \mathbb{Z}$ that satisfies $mb \leq n+1$ for a general field $K$

I am studying real algebraic geometry. Let $K$ be an Archimedean Ordered field. While trying to prove this lemma, I am not able to understand how can we choose a minimal $ n \in \mathbb{Z}$ which ...
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1answer
63 views

Show that quadratic form is anisotropic over rational numbers

How to determine if quadratic form $q(x,y,z)=-6x^2+ \frac{7}{2}y^2-\frac{25}{7}z^2$ is anisotropic over $\mathbb{Q}$? This quadratic form is a diagonalisation of another quadratic form. Is there any ...
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1answer
96 views

If $\omega \in \bigwedge^k\mathbb{R}^d$ is decomposable over $\mathbb C$, is it decomposable over $\mathbb R$?

Let $k,d$ be positive integers, and let $\omega \in \bigwedge^k\mathbb{R}^d$ be decomposable in $ \bigwedge^k\mathbb{C}^d$. Is $\omega$ decomposable in $\bigwedge^k\mathbb{R}^d$? Edit: Let me be ...
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53 views

Is every basis for $\bigwedge^kV$ satisfying a “complementary” property a rescaling of a “standard” basis?

This question was inspired by this beautiful answer: Let $V$ be a $4$-dimensional real vector space. Let $\omega_{i_1,i_2}$ ($1 \le i_1 < \ldots < i_2 \le 4$) be a basis for $\bigwedge^2V$, ...
4
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1answer
77 views

Characterising minors of diagonal matrices

Let $k,d$ be positive integers, $1<k<d$. Let $\lambda_I=\lambda_{i_1,\ldots,i_k}$ be real numbers, indexed by multi-indices $I=(i_1,\ldots,i_k)$, where $1\le i_1<\ldots<i_k \le d$. Are ...
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94 views

When do polynomial equations come from complexification?

If $f(z) \in \mathbb{C}[z]$ is a polynomial of degree $d$, then it has $d$ complex zeros. Writing the complexification $$f(x+iy)=u(x,y)+iv(x,y)$$ we observe that the real polynomial system $u(x,y)=v(x,...
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1answer
82 views

When do two plane cubic curves have 9 real intersection?

What is the "minimal" condition I can have such that two plane cubic curve defined each by one implicit equation over the reals will have 9 distinct real intersections? Note that I do not want an ...
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94 views

Concerning the ring of continuous functions on $\mathbb{R}$

It is not difficult to check that the set of continuous functions from $\mathbb{R}$ to $\mathbb{R}$ is a ring (an $\mathbb{R}$-algebra), and similarly (if I am not wrong), the set of continuous ...
2
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1answer
130 views

Denseness of algebraic points within a variety

I want to use the following statement concerning varieties, but I do not know why it is true. Claim. Let $V \subset \mathbb{C}^n$ be a variety defined over $\mathbb{Q}$. Then the set $V \cap \bar{\...
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71 views

Belonging to the same connected component of a semialgebraic set

Warm-up and main questions: Let $X \subset \mathbb{R}^n$ be a semialgebraic set and $x_1, x_2 \in X$. How can I effectively check if $x_1$ and $x_2$ are in the same connected component of $X$? Let $f ...
4
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2answers
195 views

Simultaneous real solution of $x^3+y^3+1+6xy=0$ & $xy^2+y+x^2=0$

I am trying to solve the following system of non-linear equations in real numbers: $x^3+y^3+1+6xy=0$ & $xy^2+y+x^2=0$, with $x,y$ real. I can only see that $xy\ne 0$. I have no clue whether a ...
4
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1answer
149 views

Is the minimizer of the distance from a point to a closed set generically unique?

Let $\mathcal{C} \subset \mathbb{R}^n$ be a closed set and let $E$ be the set of points in $\mathbb{R}^n$ for which there is not a unique closest element of $\mathcal{C}$. That is, if $x \in E$, then ...
2
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1answer
170 views

Algebraic geometry over general fields via Galois theory

I was informed by my supervisor that it is possible to study algebraic sets over non-algebraically closed fields by passing to an algebraically closed setting via Galois theory. How exactly do we do ...
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0answers
110 views

General techniques for proving coefficients of a multinomial are all positive

I have encountered a problem in my research where I need to prove that all the coefficients of a certain multinomial of the form $R_k(\mathbf{x}, \mathbf{y}) = \frac{P_k(\mathbf{x}, \mathbf{y})}{Q_k(\...