High Order Polynomials: Questions & Answers

[azzaz]

Hello

I am working on high order polynomials and I have two questions:1. Is there a method other than the one based on the "synthetic division with testing the signs" for identifying the lower and upper limits of the real roots?

2. Is there a way for identifying the smallest interval on the x-axis between two successive real roots, i.e. is there a theorem or test that can provide a limit on the size of the x-interval that separates successive real roots without knowing the roots?

Thank you for your help!

 

[S.G. Janssens]

I'm not sufficiently knowledgeable to answer your questions, but I suppose you have tried to relate this to a matrix eigenvalue problem? Maybe

https://en.wikipedia.org/wiki/Gershgorin_circle_theorem

and

https://en.wikipedia.org/wiki/Poincaré_separation_theorem

could provide a starting point for a search in the matrix analysis literature?

 

[mathman]

Hello

I am working on high order polynomials and I have two questions:1. Is there a method other than the one based on the "synthetic division with testing the signs" for identifying the lower and upper limits of the real roots?

2. Is there a way for identifying the smallest interval on the x-axis between two successive real roots, i.e. is there a theorem or test that can provide a limit on the size of the x-interval that separates successive real roots without knowing the roots?

Thank you for your help!

https://en.wikipedia.org/wiki/Sturm's_theorem

The above describes a method to obtain intervals to bracket individual real roots. I believe it can be adapted to answer your questions.