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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.

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.

234 questions
5
votes
1 answer

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 $f$ and $g$ be two real (nonzero) homogeneous…
Malkoun
  • 5,276
3
votes
0 answers

the number of connected components of the intersection of two algebraic sets

I have two polynomials $p$ and $q$ in $\mathbb R^n$. Is there a bound on the number of connected components of $\{x \in \mathbb R^n : p(x) = 0, q(x) \neq 0\}$ in terms of the degrees of $p$ and $q$? (For example, if $n = 2$ and $\deg p = 1$, then…
Alan C
  • 2,020
2
votes
2 answers

Identifying rings of bounded functions

Let $S$ be the subset of $\mathbb{R}^2$ defined by the inequalities $$0\le\sqrt{2}x-y<1.$$ The set of polynomials in the ring $\mathbb{R}[x,y]$ that are bounded on $S$ obviously forms a subring $B$. I suspect that $B=\mathbb{R}[\sqrt{2}x-y]$. Can…
falang
  • 559
2
votes
1 answer

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 that $V$ can have many connected components (a…
Alexandre Paiva Barreto
  • 21
2
votes
0 answers

Asymptotes of semialgebraic sets

Let $S \subseteq \mathbb{R}^n$ be an unbounded semialgebraic set. Is there a standard accepted definition of what it means for a linear-affine subspace $L$ of $\mathbb{R}^n$ to be asymptotic to $S$? The (vague) general idea is that points of $L$…
falang
  • 559
2
votes
1 answer

Sparseness of real algebraic sets

Let $f$ be a non-zero polynomial in $n$ variables with real coefficients. It seems intuitively clear that there are balls in $\mathbb{R}^n$ with arbitrarily large radius that do not meet the zero set of $f$. Is there a nice clean simple way to see…
gaddy
  • 361
1
vote
1 answer

Convex subsets of semialgebraic sets

Let's say that a semialgebraic set $S \subseteq\mathbb{R}^n$ is "thick" if there is a semialgebraic $S_1 \subseteq \mathbb{R}^n$ and a positive $\epsilon$ such that $$\bigcup_{p\in S_1}B(p,\epsilon) = S.$$ Here $B(p,\epsilon)$ is the ball in…
gaddy
  • 361
1
vote
1 answer

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 satisfies $mb \leq n+1$. $K$ being an Archimedean…
MUH
  • 1,377
1
vote
0 answers

Negative homogenization on the positive orthant

I am trying to find a polynomial over $\mathbb{R}$ that is strictly positive on the positive orthant but its homogenization is not (except at 0). I d especially like to know if there is an example in 1 variable. The question is probably not that…
user336083
  • 11
1
vote
1 answer

how to prove that a set is not semi-algebraic

I've been reading about semi-algebraic sets and I've run into several examples of sets which have been proved to not be semi-algebraic. Examples include the integers, and the curves of the sine and exponential functions. I'm really curious about…
Joel Turnblade
  • 353
1
vote
1 answer

Non-negative polynomials and sums of squares

Let $\mathcal{P}_{n,2d}$ denote the real forms of degree $2d$ in $n$ variables and $\Sigma_{n,2d}$ the forms that can be written as sums of squares. The Motzkin-polynomial shows that $\mathcal{P}_{3,6}\neq\Sigma_{3,6}$. Starting with this…
Jeff
  • 21
0
votes
0 answers

When is a solution to a multivariate polynomial not parameterizable?

This is a follow up to a comment made on one of my previous questions that I feel makes more sense as a separate question. The comment was to the effect that the real solutions to a system of polynomials may not be parameterizable unless the system…
Michael Stachowsky
  • 2,943
0
votes
2 answers

Expression with Different Units

So I've tried to solve this problem by myself but I'm not sure I'm right. So here it goes: In baseball, the pitcher's mound is 60.5 feet from home plate. The strike zone, or distance across the plate, is 17 inches. The time it takes for a baseball…
Marisol
  • 15