Bounds of roots for a parametric quartic equation
Clash Royale CLAN TAG#URR8PPP
up vote
0
down vote
favorite
I have the following quartic equation
$$omega_4 x^4+omega_3 x^3+omega_2 x^2+omega_1 x+omega_0=0$$
where $omega_i$ depend on several real parameters. I'm not interested in searching its roots, but I would like to respond to the following point:
- What are the bounds (lower and upper) for the absolute value of all the roots (real or complex)?
polynomials
add a comment |Â
up vote
0
down vote
favorite
I have the following quartic equation
$$omega_4 x^4+omega_3 x^3+omega_2 x^2+omega_1 x+omega_0=0$$
where $omega_i$ depend on several real parameters. I'm not interested in searching its roots, but I would like to respond to the following point:
- What are the bounds (lower and upper) for the absolute value of all the roots (real or complex)?
polynomials
Possible duplicate: math.stackexchange.com/questions/1359009/â¦
â A. Pongrácz
Aug 16 at 5:13
2
Possible duplicate of How to count the real roots of a quartic equation?
â A. Pongrácz
Aug 16 at 5:14
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I have the following quartic equation
$$omega_4 x^4+omega_3 x^3+omega_2 x^2+omega_1 x+omega_0=0$$
where $omega_i$ depend on several real parameters. I'm not interested in searching its roots, but I would like to respond to the following point:
- What are the bounds (lower and upper) for the absolute value of all the roots (real or complex)?
polynomials
I have the following quartic equation
$$omega_4 x^4+omega_3 x^3+omega_2 x^2+omega_1 x+omega_0=0$$
where $omega_i$ depend on several real parameters. I'm not interested in searching its roots, but I would like to respond to the following point:
- What are the bounds (lower and upper) for the absolute value of all the roots (real or complex)?
polynomials
edited Aug 16 at 16:56
asked Aug 16 at 5:06
Mark
3,56751746
3,56751746
Possible duplicate: math.stackexchange.com/questions/1359009/â¦
â A. Pongrácz
Aug 16 at 5:13
2
Possible duplicate of How to count the real roots of a quartic equation?
â A. Pongrácz
Aug 16 at 5:14
add a comment |Â
Possible duplicate: math.stackexchange.com/questions/1359009/â¦
â A. Pongrácz
Aug 16 at 5:13
2
Possible duplicate of How to count the real roots of a quartic equation?
â A. Pongrácz
Aug 16 at 5:14
Possible duplicate: math.stackexchange.com/questions/1359009/â¦
â A. Pongrácz
Aug 16 at 5:13
Possible duplicate: math.stackexchange.com/questions/1359009/â¦
â A. Pongrácz
Aug 16 at 5:13
2
2
Possible duplicate of How to count the real roots of a quartic equation?
â A. Pongrácz
Aug 16 at 5:14
Possible duplicate of How to count the real roots of a quartic equation?
â A. Pongrácz
Aug 16 at 5:14
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
0
down vote
There is some radius $r$ such that when $|x|>r,$
$$|omega_4 x^4| > |omega_3 x^3| + |omega_2 x^2| + |omega_1 x| + |omega_0| > |omega_3 x^3 + omega_2 x^2 + omega_1 x + omega_0| $$
Rouche's theorem says that when this is the case, all of the roots will be within a disk of radius $r.$
Thank you. What is $z$? If all the roots are real, this means that they will be in the range $[-x,x]$?
â Mark
Aug 17 at 7:09
This is a proof of complex analysis where polynomials are usually expressed in terms of $z.$ But, as I have not introduced z, I will change it.
â Doug M
Aug 17 at 16:39
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
There is some radius $r$ such that when $|x|>r,$
$$|omega_4 x^4| > |omega_3 x^3| + |omega_2 x^2| + |omega_1 x| + |omega_0| > |omega_3 x^3 + omega_2 x^2 + omega_1 x + omega_0| $$
Rouche's theorem says that when this is the case, all of the roots will be within a disk of radius $r.$
Thank you. What is $z$? If all the roots are real, this means that they will be in the range $[-x,x]$?
â Mark
Aug 17 at 7:09
This is a proof of complex analysis where polynomials are usually expressed in terms of $z.$ But, as I have not introduced z, I will change it.
â Doug M
Aug 17 at 16:39
add a comment |Â
up vote
0
down vote
There is some radius $r$ such that when $|x|>r,$
$$|omega_4 x^4| > |omega_3 x^3| + |omega_2 x^2| + |omega_1 x| + |omega_0| > |omega_3 x^3 + omega_2 x^2 + omega_1 x + omega_0| $$
Rouche's theorem says that when this is the case, all of the roots will be within a disk of radius $r.$
Thank you. What is $z$? If all the roots are real, this means that they will be in the range $[-x,x]$?
â Mark
Aug 17 at 7:09
This is a proof of complex analysis where polynomials are usually expressed in terms of $z.$ But, as I have not introduced z, I will change it.
â Doug M
Aug 17 at 16:39
add a comment |Â
up vote
0
down vote
up vote
0
down vote
There is some radius $r$ such that when $|x|>r,$
$$|omega_4 x^4| > |omega_3 x^3| + |omega_2 x^2| + |omega_1 x| + |omega_0| > |omega_3 x^3 + omega_2 x^2 + omega_1 x + omega_0| $$
Rouche's theorem says that when this is the case, all of the roots will be within a disk of radius $r.$
There is some radius $r$ such that when $|x|>r,$
$$|omega_4 x^4| > |omega_3 x^3| + |omega_2 x^2| + |omega_1 x| + |omega_0| > |omega_3 x^3 + omega_2 x^2 + omega_1 x + omega_0| $$
Rouche's theorem says that when this is the case, all of the roots will be within a disk of radius $r.$
edited Aug 17 at 16:40
answered Aug 16 at 18:32
Doug M
39.3k31749
39.3k31749
Thank you. What is $z$? If all the roots are real, this means that they will be in the range $[-x,x]$?
â Mark
Aug 17 at 7:09
This is a proof of complex analysis where polynomials are usually expressed in terms of $z.$ But, as I have not introduced z, I will change it.
â Doug M
Aug 17 at 16:39
add a comment |Â
Thank you. What is $z$? If all the roots are real, this means that they will be in the range $[-x,x]$?
â Mark
Aug 17 at 7:09
This is a proof of complex analysis where polynomials are usually expressed in terms of $z.$ But, as I have not introduced z, I will change it.
â Doug M
Aug 17 at 16:39
Thank you. What is $z$? If all the roots are real, this means that they will be in the range $[-x,x]$?
â Mark
Aug 17 at 7:09
Thank you. What is $z$? If all the roots are real, this means that they will be in the range $[-x,x]$?
â Mark
Aug 17 at 7:09
This is a proof of complex analysis where polynomials are usually expressed in terms of $z.$ But, as I have not introduced z, I will change it.
â Doug M
Aug 17 at 16:39
This is a proof of complex analysis where polynomials are usually expressed in terms of $z.$ But, as I have not introduced z, I will change it.
â Doug M
Aug 17 at 16:39
add a comment |Â
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2884423%2fbounds-of-roots-for-a-parametric-quartic-equation%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Possible duplicate: math.stackexchange.com/questions/1359009/â¦
â A. Pongrácz
Aug 16 at 5:13
2
Possible duplicate of How to count the real roots of a quartic equation?
â A. Pongrácz
Aug 16 at 5:14