Is the set of real coefficients of monic polynomial with roots in the unit open disk regular open?
$begingroup$
Let $pi: mathbb R^n to mathcal P_n$ denote the bijection between the coefficients and the real monic $n^{text{th}}$ degree polynomials. That is, for $a = (a_0, dots, a_{n-1}) in mathbb R^n$, $pi(a) = x^n + a_{n-1} x^{n-1} + dots + a_0$.
Let us denote
$$ Delta = { v in mathbb R^n: pi(v) text{ has all roots in the open unit disk of } mathbb C},$$
and
$$ Gamma = { v in mathbb R^n: pi(v) text{ has all roots in the closed unit disk of } mathbb C}.$$
I think we can show $Delta$ is open and $Gamma$ is closed in Eclidean topology. With Vieta's formula, we can also claim $Delta$ and $Gamma$ are both path connected. If I am not mistaken, we can also show $overline{Delta} = Gamma$ where $bar{cdot}$ denotes closure of a set. Because for $w in Gamma setminus Delta$ with roots $r_1, dots, r_n in mathbb C$, then
begin{align*}
u(alpha) := pi^{-1}left( (x-alpha r_1) dots (x-alpha r_n) right)
end{align*}
is in $Delta$ when $alpha < 1$ and $lim_{alpha to 1} u(alpha) = w$.
My question is: can we claim $Delta$ is regular open, that is the interior of $Gamma$ is $Delta$, i.e., $Gamma^{circ} = Delta$?
linear-algebra abstract-algebra general-topology polynomials
asked Dec 4 '18 at 2:46
user1101010user1101010
8121730
$endgroup$
add a comment |
$begingroup$
Let $pi: mathbb R^n to mathcal P_n$ denote the bijection between the coefficients and the real monic $n^{text{th}}$ degree polynomials. That is, for $a = (a_0, dots, a_{n-1}) in mathbb R^n$, $pi(a) = x^n + a_{n-1} x^{n-1} + dots + a_0$.
Let us denote
$$ Delta = { v in mathbb R^n: pi(v) text{ has all roots in the open unit disk of } mathbb C},$$
and
$$ Gamma = { v in mathbb R^n: pi(v) text{ has all roots in the closed unit disk of } mathbb C}.$$
I think we can show $Delta$ is open and $Gamma$ is closed in Eclidean topology. With Vieta's formula, we can also claim $Delta$ and $Gamma$ are both path connected. If I am not mistaken, we can also show $overline{Delta} = Gamma$ where $bar{cdot}$ denotes closure of a set. Because for $w in Gamma setminus Delta$ with roots $r_1, dots, r_n in mathbb C$, then
begin{align*}
u(alpha) := pi^{-1}left( (x-alpha r_1) dots (x-alpha r_n) right)
end{align*}
is in $Delta$ when $alpha < 1$ and $lim_{alpha to 1} u(alpha) = w$.
My question is: can we claim $Delta$ is regular open, that is the interior of $Gamma$ is $Delta$, i.e., $Gamma^{circ} = Delta$?
linear-algebra abstract-algebra general-topology polynomials
asked Dec 4 '18 at 2:46
user1101010user1101010
8121730
$endgroup$
$begingroup$
I'll try and fix my answer.
$endgroup$
– Not Mike
Dec 4 '18 at 7:03
add a comment |
$begingroup$
Let $pi: mathbb R^n to mathcal P_n$ denote the bijection between the coefficients and the real monic $n^{text{th}}$ degree polynomials. That is, for $a = (a_0, dots, a_{n-1}) in mathbb R^n$, $pi(a) = x^n + a_{n-1} x^{n-1} + dots + a_0$.
Let us denote
$$ Delta = { v in mathbb R^n: pi(v) text{ has all roots in the open unit disk of } mathbb C},$$
and
$$ Gamma = { v in mathbb R^n: pi(v) text{ has all roots in the closed unit disk of } mathbb C}.$$
I think we can show $Delta$ is open and $Gamma$ is closed in Eclidean topology. With Vieta's formula, we can also claim $Delta$ and $Gamma$ are both path connected. If I am not mistaken, we can also show $overline{Delta} = Gamma$ where $bar{cdot}$ denotes closure of a set. Because for $w in Gamma setminus Delta$ with roots $r_1, dots, r_n in mathbb C$, then
begin{align*}
u(alpha) := pi^{-1}left( (x-alpha r_1) dots (x-alpha r_n) right)
end{align*}
is in $Delta$ when $alpha < 1$ and $lim_{alpha to 1} u(alpha) = w$.
My question is: can we claim $Delta$ is regular open, that is the interior of $Gamma$ is $Delta$, i.e., $Gamma^{circ} = Delta$?
linear-algebra abstract-algebra general-topology polynomials
asked Dec 4 '18 at 2:46
user1101010user1101010
8121730
$endgroup$
Let $pi: mathbb R^n to mathcal P_n$ denote the bijection between the coefficients and the real monic $n^{text{th}}$ degree polynomials. That is, for $a = (a_0, dots, a_{n-1}) in mathbb R^n$, $pi(a) = x^n + a_{n-1} x^{n-1} + dots + a_0$.
Let us denote
$$ Delta = { v in mathbb R^n: pi(v) text{ has all roots in the open unit disk of } mathbb C},$$
and
$$ Gamma = { v in mathbb R^n: pi(v) text{ has all roots in the closed unit disk of } mathbb C}.$$
I think we can show $Delta$ is open and $Gamma$ is closed in Eclidean topology. With Vieta's formula, we can also claim $Delta$ and $Gamma$ are both path connected. If I am not mistaken, we can also show $overline{Delta} = Gamma$ where $bar{cdot}$ denotes closure of a set. Because for $w in Gamma setminus Delta$ with roots $r_1, dots, r_n in mathbb C$, then
begin{align*}
u(alpha) := pi^{-1}left( (x-alpha r_1) dots (x-alpha r_n) right)
end{align*}
is in $Delta$ when $alpha < 1$ and $lim_{alpha to 1} u(alpha) = w$.
My question is: can we claim $Delta$ is regular open, that is the interior of $Gamma$ is $Delta$, i.e., $Gamma^{circ} = Delta$?
linear-algebra abstract-algebra general-topology polynomials
linear-algebra abstract-algebra general-topology polynomials
asked Dec 4 '18 at 2:46
user1101010user1101010
8121730
asked Dec 4 '18 at 2:46
user1101010user1101010
8121730
edited Dec 4 '18 at 19:57
user1101010
asked Dec 4 '18 at 2:46
user1101010user1101010
8121730
asked Dec 4 '18 at 2:46
user1101010user1101010
8121730
asked Dec 4 '18 at 2:46
user1101010user1101010
8121730
8121730
$begingroup$
I'll try and fix my answer.
$endgroup$
– Not Mike
Dec 4 '18 at 7:03
add a comment |
$begingroup$
I'll try and fix my answer.
$endgroup$
– Not Mike
Dec 4 '18 at 7:03
$begingroup$
I'll try and fix my answer.
$endgroup$
– Not Mike
Dec 4 '18 at 7:03
$begingroup$
I'll try and fix my answer.
$endgroup$
– Not Mike
Dec 4 '18 at 7:03
add a comment |
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$begingroup$
I'll try and fix my answer.
$endgroup$
– Not Mike
Dec 4 '18 at 7:03