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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Checking if a polynomial ideal is real

I'm working with a polynomial ideal $I \subset \mathbb{R}[s_1,c_1, \ldots, s_n, c_n]$ generated as $I = \langle s_1^2+c_1^2-1, \ldots, s_n^2+c_n^2-1\rangle$ and looking to show that this ideal is real....
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1 vote
31 views

Polynomial formula for orthogonal vector in odd dimensions [duplicate]

I have been thinking about this problem recently. In 2 dimensions there is an easy formula for a nonzero vector orthogonal to a given vector $(x, y)$, namely $(-y, x)$. By taking pairs of coordinates, ...
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64 views

Partial derivatives "split" over $\mathbb R$

Kaplansky's book Fields and Rings, page 30, implicitly contains the following question. For which fields $K$ does the following statement hold true for every polynomial $g\in K[X]$: If $g$ splits ...
• 1,780
66 views

A semi-algebraic set in $\mathbb{R}^d$ has dimension $d$ if and only if its interior is non empty

I would like to prove the following result : A semi-algebraic set in $\mathbb{R}^d$ has dimension $d$ if and only if its interior is non empty The dimension here has to be understood in the semi ...
• 1,381
113 views

Can we determine into how many regions the zero set of a polynomial divides the plane?

Disclaimer: If you look at my other questions, you will see that I am still learning the basics of algebra. I am well aware that the answer to this question is likely to go beyond my current level of ...
• 695
14 views

Example of semi algebraic subset over a real closed field

I am trying to find interesting example of semi algebraic subset over a real closed field other than those we have by considering $\mathbb{R}^n$. I tried to construct an example involving matrices. ...
• 1,381
1 vote
36 views

Semi algebraic sets and smooth manifolds

I would like to prove a result that I cannot see why it could be wrong in order to fix the idea. Here is the result : Let $A\subset\mathbb{R}^{n}$ be a semi algebraic set which is a smooth ...
• 1,381
1 vote
34 views

Link between semi-algebraic dimension and vector space dimension

I am new to semi-algebraic geometry and I have encountered a very pleasant proposition to define the semi-algebraic dimension of a semi-algebraic subset $A$ of $\mathbb{R}^n$. This definition is based ...
• 1,381
29 views

Difficulty to prove that a set is semi algebraic

I consider for $i\in\{1,…,N\}$ the subset $X_i\subset\mathbb{R}^{n}$ that are convex and compact and semi algebraic. We denote by $X=\Pi_{i}^{N}X_i$ the Cartesian product of the $X_i$ which is still ...
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rational map by F. Mangolte

I'm reading Real Algebraic Varieties by F. Mangolte. Definition 1.3.22 (in the book) If $X$ and $Y$ are algebraic varieties over a base field $K$ a rational map $\phi:X\dashrightarrow Y$ is an ...
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1 vote
47 views

misunderstanding on real algebraic varieties

Bochnak-Coste-Roy's book "Real Algebraic Geometry" (1998) is probably the main reference on this subject. I am probably misunderstanding something very fundamental as I can apparently find ...
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Notation $\text{RSpec}(K)$ for space of orderings of a field $K$

I'm reading notes from commutative algebra by Pete Clark. In one of the examples of Galois connections he introduces notation $\text{RSpec}(K)$ for the set of all total orders on a field $K$, and ...
• 10.5k
1 vote
75 views

Semi-numeric solutions to a system of polynomial equations when a Groebner basis is too complicated

I have a system of polynomial equations with rational coefficients and I would like to find real solutions, if they exist. The system has $n\sim 10$ unknowns, $n$ equations with degree $\sim 2n$ and ...
• 4,886
1 vote
41 views

Homotopy equivalence vs deformation retract for analytic spaces

I know that there are many examples of spaces $Y\subset X$ such that $X$ and $Y$ are homotopy equivalent but there is no deformation retract of $X$ to $Y$ (e.g., Does homotopy equivalence to a ...
• 404
1 vote
33 views

Connected components of the zero set of multilinear polynomials

Let $p\in \mathbb{R}[x_1, \ldots, x_n]$ be a real (non-constant) multilinear polynomial of degree $\deg p=d$, then its real zero-set $V_{\mathbb{R}}(p)$ is non-empty and has dimension $n-1$ by the ...
• 53
68 views

Proving a system of quadratic forms has no (non-zero) solutions

A system of homogeneous linear equations always has the solution $x=(0,\dots, 0)$. Suppose we have a system of $n$ homogeneous linear equations in $k$ variables. If $k > n$ then there will ...
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1 vote
51 views

System of $n$ simultaneously diagonal real quadrics in $n+1$ variables has all solutions real

A single isotropic real quadric in 2 variables always has 2 (projective) real solutions. See Zero set of system of two real quadratic forms for the explicit form. I've noticed that a system of 2 ...
• 8,552
1 vote
58 views

When are the zeroes from Bezout's theorem real/rational/integral

Suppose I have $n$ quadratic homogeneous polynomials $f_1, \cdots, f_n$ in $n+1$ variables over the field $\mathbb{C}$. Bezout's theorem says that generically there will be $2^n$ common zeroes. I was ...
1 vote
45 views

Help me complete a proof about the enumeration of similar intersections of an algebraic surface with a hyperplane

This is something of a follow-up to this question that i posted almost a year ago: What are the conditions on two multivariate real polynomials $f$ and $g$ so that their zero sets are similar? It was ...
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Question on matrix spaces that is derived from several vector spaces

Suppose I have $m$ column vectors $x_i \in X \subset \mathbb{R}^n$, $i \in \{1,\ldots,n\}$. If I construct a matrix $\mathbf{x}= [x_1^T ; \ldots ; x_m^T]$, then what set does $\mathbf{x}$ live in ...
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1 vote
42 views

What is the minimal number of polynomial squares needed to represent any sum of any number of polynomial squares?

Let $Q^{(n, d)}$ be a set of real polynomials in $n$ variables of degree $d$, and $S_k^{(n, d)}$ the set of the sum of squares of any $k$ such polynomials: S_k^{(n, d)} := \left\...
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101 views

Proof of uniqueness of real-closure of an ordered field

I'm reading the proof of uniqueness of real-closure of an ordered field $F$, that is an algebraic extension $R$ of $F$ such that $R$ is real-closed and the unique order on $R$ extends that of $F$. I ...
• 10.5k
29 views

Theorems of calculus for real-closed fields [duplicate]

Theorems such as Intermediate value theorem Rolle's theorem Mean value theorem Positive/negative derivative implies strictly increasing/decreasing Hold for polynomials over real-closed fields. If ...
• 10.5k
1 vote
60 views

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