Difference between revisions of "Descartes theorem"
From Encyclopedia of Mathematics
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| − | A theorem according to which the number of positive roots of a polynomial with real coefficients is equal to, or is an even number smaller than, the number of changes of sign in the series of its coefficients (each root being counted the number of times equal to its multiplicity); zero coefficients are disregarded in counting changes of sign. If it is known that all roots of a given polynomial are real (e.g. for the characteristic polynomial of a symmetric matrix), Descartes' theorem yields the exact number of roots. This theorem can also be used to find the number of negative roots of a polynomial | + | A theorem according to which the number of positive roots of a polynomial with real coefficients is equal to, or is an even number smaller than, the number of changes of sign in the series of its coefficients (each root being counted the number of times equal to its multiplicity); zero coefficients are disregarded in counting changes of sign. If it is known that all roots of a given polynomial are real (e.g. for the characteristic polynomial of a symmetric matrix), Descartes' theorem yields the exact number of roots. This theorem can also be used to find the number of negative roots of a polynomial $f(x)$ by considering $f(-x)$. |
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| − | + | |valign="top|{{Ref|Ku}}||valign="top"| A.G. Kurosh, "Higher algebra" , MIR (1972) (Translated from Russian) | |
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| − | + | |valign="top"|{{Ref|Ko}}||valign="top"| A.I. Kostrikin, "Introduction to algebra" , Springer (1982) (Translated from Russian) | |
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Revision as of 10:05, 28 April 2012
Descartes sign rule
A theorem according to which the number of positive roots of a polynomial with real coefficients is equal to, or is an even number smaller than, the number of changes of sign in the series of its coefficients (each root being counted the number of times equal to its multiplicity); zero coefficients are disregarded in counting changes of sign. If it is known that all roots of a given polynomial are real (e.g. for the characteristic polynomial of a symmetric matrix), Descartes' theorem yields the exact number of roots. This theorem can also be used to find the number of negative roots of a polynomial $f(x)$ by considering $f(-x)$.
References
| [Ku] | A.G. Kurosh, "Higher algebra" , MIR (1972) (Translated from Russian) |
| [Ko] | A.I. Kostrikin, "Introduction to algebra" , Springer (1982) (Translated from Russian) |
How to Cite This Entry:
Descartes theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Descartes_theorem&oldid=12409
Descartes theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Descartes_theorem&oldid=12409
This article was adapted from an original article by I.V. Proskuryakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article