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 $f in mathbb{R}[x]$ be a multivariable polynomial. Let $X_1, X_2 subset mathbb{R}^n$ be an algebraic curves with parametrisation at infinity $g_1, g_2$ respectivly ($g_1:[0,infty)to X_1$, $lim_{ttoinfty}|g_1(t)|=+infty $, $g_2:[0,infty)to X_2$, $lim_{ttoinfty}|g_2(t)|=+infty $ ). Assume that
$$lim_{ttoinfty}(f(g_1(t))=lim_{ttoinfty}(f(g_2(t))=0.$$
How can I check if $g_1([R,infty))$ and $g_2([R,infty))$ are in the same connected component of the set $f^{-1}((-varepsilon, varepsilon))$ (for $varepsilon>0$ small enough, $R>0$ big enough)?
algebraic-topology connectedness real-algebraic-geometry semialgebraic-geometry
edited Dec 8 at 16:59
Rodrigo de Azevedo
12.8k41854
asked Nov 13 at 23:40
hakunamatata
213
add a comment |
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 $f in mathbb{R}[x]$ be a multivariable polynomial. Let $X_1, X_2 subset mathbb{R}^n$ be an algebraic curves with parametrisation at infinity $g_1, g_2$ respectivly ($g_1:[0,infty)to X_1$, $lim_{ttoinfty}|g_1(t)|=+infty $, $g_2:[0,infty)to X_2$, $lim_{ttoinfty}|g_2(t)|=+infty $ ). Assume that
$$lim_{ttoinfty}(f(g_1(t))=lim_{ttoinfty}(f(g_2(t))=0.$$
How can I check if $g_1([R,infty))$ and $g_2([R,infty))$ are in the same connected component of the set $f^{-1}((-varepsilon, varepsilon))$ (for $varepsilon>0$ small enough, $R>0$ big enough)?
algebraic-topology connectedness real-algebraic-geometry semialgebraic-geometry
edited Dec 8 at 16:59
Rodrigo de Azevedo
12.8k41854
asked Nov 13 at 23:40
hakunamatata
213
Take a look at chapter 16 of Algorithms in Real Algebraic Geometry [PDF].
– Rodrigo de Azevedo
Dec 8 at 20:37
Thank you for the replay. Could you give me some more details how to attempt this problems? I'm not familliar with the book and it's seems overwhelming at first.
– hakunamatata
Dec 13 at 19:47
It's overwhelming to me, too.
– Rodrigo de Azevedo
Dec 14 at 1:22
add a comment |
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 $f in mathbb{R}[x]$ be a multivariable polynomial. Let $X_1, X_2 subset mathbb{R}^n$ be an algebraic curves with parametrisation at infinity $g_1, g_2$ respectivly ($g_1:[0,infty)to X_1$, $lim_{ttoinfty}|g_1(t)|=+infty $, $g_2:[0,infty)to X_2$, $lim_{ttoinfty}|g_2(t)|=+infty $ ). Assume that
$$lim_{ttoinfty}(f(g_1(t))=lim_{ttoinfty}(f(g_2(t))=0.$$
How can I check if $g_1([R,infty))$ and $g_2([R,infty))$ are in the same connected component of the set $f^{-1}((-varepsilon, varepsilon))$ (for $varepsilon>0$ small enough, $R>0$ big enough)?
algebraic-topology connectedness real-algebraic-geometry semialgebraic-geometry
edited Dec 8 at 16:59
Rodrigo de Azevedo
12.8k41854
asked Nov 13 at 23:40
hakunamatata
213
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 $f in mathbb{R}[x]$ be a multivariable polynomial. Let $X_1, X_2 subset mathbb{R}^n$ be an algebraic curves with parametrisation at infinity $g_1, g_2$ respectivly ($g_1:[0,infty)to X_1$, $lim_{ttoinfty}|g_1(t)|=+infty $, $g_2:[0,infty)to X_2$, $lim_{ttoinfty}|g_2(t)|=+infty $ ). Assume that
$$lim_{ttoinfty}(f(g_1(t))=lim_{ttoinfty}(f(g_2(t))=0.$$
How can I check if $g_1([R,infty))$ and $g_2([R,infty))$ are in the same connected component of the set $f^{-1}((-varepsilon, varepsilon))$ (for $varepsilon>0$ small enough, $R>0$ big enough)?
algebraic-topology connectedness real-algebraic-geometry semialgebraic-geometry
algebraic-topology connectedness real-algebraic-geometry semialgebraic-geometry
edited Dec 8 at 16:59
Rodrigo de Azevedo
12.8k41854
asked Nov 13 at 23:40
hakunamatata
213
edited Dec 8 at 16:59
Rodrigo de Azevedo
12.8k41854
asked Nov 13 at 23:40
hakunamatata
213
edited Dec 8 at 16:59
Rodrigo de Azevedo
12.8k41854
edited Dec 8 at 16:59
Rodrigo de Azevedo
12.8k41854
edited Dec 8 at 16:59
Rodrigo de Azevedo
12.8k41854
12.8k41854
asked Nov 13 at 23:40
hakunamatata
213
asked Nov 13 at 23:40
hakunamatata
213
asked Nov 13 at 23:40
hakunamatata
213
213
Take a look at chapter 16 of Algorithms in Real Algebraic Geometry [PDF].
– Rodrigo de Azevedo
Dec 8 at 20:37
Thank you for the replay. Could you give me some more details how to attempt this problems? I'm not familliar with the book and it's seems overwhelming at first.
– hakunamatata
Dec 13 at 19:47
It's overwhelming to me, too.
– Rodrigo de Azevedo
Dec 14 at 1:22
add a comment |
Take a look at chapter 16 of Algorithms in Real Algebraic Geometry [PDF].
– Rodrigo de Azevedo
Dec 8 at 20:37
Thank you for the replay. Could you give me some more details how to attempt this problems? I'm not familliar with the book and it's seems overwhelming at first.
– hakunamatata
Dec 13 at 19:47
It's overwhelming to me, too.
– Rodrigo de Azevedo
Dec 14 at 1:22
Take a look at chapter 16 of Algorithms in Real Algebraic Geometry [PDF].
– Rodrigo de Azevedo
Dec 8 at 20:37
Take a look at chapter 16 of Algorithms in Real Algebraic Geometry [PDF].
– Rodrigo de Azevedo
Dec 8 at 20:37
Thank you for the replay. Could you give me some more details how to attempt this problems? I'm not familliar with the book and it's seems overwhelming at first.
– hakunamatata
Dec 13 at 19:47
Thank you for the replay. Could you give me some more details how to attempt this problems? I'm not familliar with the book and it's seems overwhelming at first.
– hakunamatata
Dec 13 at 19:47
It's overwhelming to me, too.
– Rodrigo de Azevedo
Dec 14 at 1:22
It's overwhelming to me, too.
– Rodrigo de Azevedo
Dec 14 at 1:22
add a comment |
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Take a look at chapter 16 of Algorithms in Real Algebraic Geometry [PDF].
– Rodrigo de Azevedo
Dec 8 at 20:37
Thank you for the replay. Could you give me some more details how to attempt this problems? I'm not familliar with the book and it's seems overwhelming at first.
– hakunamatata
Dec 13 at 19:47
It's overwhelming to me, too.
– Rodrigo de Azevedo
Dec 14 at 1:22