Bounds of roots for a parametric quartic equation

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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:



  1. What are the bounds (lower and upper) for the absolute value of all the roots (real or complex)?



polynomials




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  • Possible duplicate: math.stackexchange.com/questions/1359009/…
    – A. Pongrácz
    Aug 16 at 5:13






  • 2




    Possible duplicate of How to count the real roots of a quartic equation?
    – A. Pongrácz
    Aug 16 at 5:14














up vote
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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:



  1. What are the bounds (lower and upper) for the absolute value of all the roots (real or complex)?



polynomials




share|cite|improve this question






















  • Possible duplicate: math.stackexchange.com/questions/1359009/…
    – A. Pongrácz
    Aug 16 at 5:13






  • 2




    Possible duplicate of How to count the real roots of a quartic equation?
    – A. Pongrácz
    Aug 16 at 5:14












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:



  1. What are the bounds (lower and upper) for the absolute value of all the roots (real or complex)?



polynomials




share|cite|improve this question














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:



  1. What are the bounds (lower and upper) for the absolute value of all the roots (real or complex)?




polynomials





share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Aug 16 at 16:56

























asked Aug 16 at 5:06









Mark

3,56751746




3,56751746











  • Possible duplicate: math.stackexchange.com/questions/1359009/…
    – A. Pongrácz
    Aug 16 at 5:13






  • 2




    Possible duplicate of How to count the real roots of a quartic equation?
    – A. Pongrácz
    Aug 16 at 5:14
















  • Possible duplicate: math.stackexchange.com/questions/1359009/…
    – A. Pongrácz
    Aug 16 at 5:13






  • 2




    Possible duplicate of How to count the real roots of a quartic equation?
    – A. Pongrácz
    Aug 16 at 5:14















Possible duplicate: math.stackexchange.com/questions/1359009/…
– A. Pongrácz
Aug 16 at 5:13




Possible duplicate: math.stackexchange.com/questions/1359009/…
– A. Pongrácz
Aug 16 at 5:13




2




2




Possible duplicate of How to count the real roots of a quartic equation?
– A. Pongrácz
Aug 16 at 5:14




Possible duplicate of How to count the real roots of a quartic equation?
– A. Pongrácz
Aug 16 at 5:14










1 Answer
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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 $r.$






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  • Thank you. What is $z$? If all the roots are real, this means that they will be in the range $[-x,x]$?
    – 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.
    – Doug M
    Aug 17 at 16:39










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1 Answer
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1 Answer
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active

oldest

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active

oldest

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oldest

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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 $r.$






share|cite|improve this answer






















  • Thank you. What is $z$? If all the roots are real, this means that they will be in the range $[-x,x]$?
    – 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.
    – Doug M
    Aug 17 at 16:39














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 $r.$






share|cite|improve this answer






















  • Thank you. What is $z$? If all the roots are real, this means that they will be in the range $[-x,x]$?
    – 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.
    – Doug M
    Aug 17 at 16:39












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 $r.$






share|cite|improve this answer














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 $r.$







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








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 $[-x,x]$?
    – 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.
    – 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 $[-x,x]$?
    – 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.
    – 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 $[-x,x]$?
– 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 $[-x,x]$?
– 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.
– 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.
– Doug M
Aug 17 at 16:39












 

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