Belonging to the same connected component of a semialgebraic set












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Warm-up and main questions:




  1. 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 $X$?


  2. 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)?











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$endgroup$












  • $begingroup$
    Take a look at chapter 16 of Algorithms in Real Algebraic Geometry [PDF].
    $endgroup$
    – Rodrigo de Azevedo
    Dec 8 '18 at 20:37










  • $begingroup$
    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.
    $endgroup$
    – hakunamatata
    Dec 13 '18 at 19:47










  • $begingroup$
    It's overwhelming to me, too.
    $endgroup$
    – Rodrigo de Azevedo
    Dec 14 '18 at 1:22
















1












$begingroup$


Warm-up and main questions:




  1. 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 $X$?


  2. 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)?











share|cite|improve this question











$endgroup$












  • $begingroup$
    Take a look at chapter 16 of Algorithms in Real Algebraic Geometry [PDF].
    $endgroup$
    – Rodrigo de Azevedo
    Dec 8 '18 at 20:37










  • $begingroup$
    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.
    $endgroup$
    – hakunamatata
    Dec 13 '18 at 19:47










  • $begingroup$
    It's overwhelming to me, too.
    $endgroup$
    – Rodrigo de Azevedo
    Dec 14 '18 at 1:22














1












1








1





$begingroup$


Warm-up and main questions:




  1. 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 $X$?


  2. 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)?











share|cite|improve this question











$endgroup$




Warm-up and main questions:




  1. 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 $X$?


  2. 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






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edited Dec 8 '18 at 16:59









Rodrigo de Azevedo

13k41958




13k41958










asked Nov 13 '18 at 23:40









hakunamatatahakunamatata

234




234












  • $begingroup$
    Take a look at chapter 16 of Algorithms in Real Algebraic Geometry [PDF].
    $endgroup$
    – Rodrigo de Azevedo
    Dec 8 '18 at 20:37










  • $begingroup$
    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.
    $endgroup$
    – hakunamatata
    Dec 13 '18 at 19:47










  • $begingroup$
    It's overwhelming to me, too.
    $endgroup$
    – Rodrigo de Azevedo
    Dec 14 '18 at 1:22


















  • $begingroup$
    Take a look at chapter 16 of Algorithms in Real Algebraic Geometry [PDF].
    $endgroup$
    – Rodrigo de Azevedo
    Dec 8 '18 at 20:37










  • $begingroup$
    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.
    $endgroup$
    – hakunamatata
    Dec 13 '18 at 19:47










  • $begingroup$
    It's overwhelming to me, too.
    $endgroup$
    – Rodrigo de Azevedo
    Dec 14 '18 at 1:22
















$begingroup$
Take a look at chapter 16 of Algorithms in Real Algebraic Geometry [PDF].
$endgroup$
– Rodrigo de Azevedo
Dec 8 '18 at 20:37




$begingroup$
Take a look at chapter 16 of Algorithms in Real Algebraic Geometry [PDF].
$endgroup$
– Rodrigo de Azevedo
Dec 8 '18 at 20:37












$begingroup$
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.
$endgroup$
– hakunamatata
Dec 13 '18 at 19:47




$begingroup$
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.
$endgroup$
– hakunamatata
Dec 13 '18 at 19:47












$begingroup$
It's overwhelming to me, too.
$endgroup$
– Rodrigo de Azevedo
Dec 14 '18 at 1:22




$begingroup$
It's overwhelming to me, too.
$endgroup$
– Rodrigo de Azevedo
Dec 14 '18 at 1:22










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