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Counting zeros of generalised polynomials: Descartes’ rule of signs and Laguerre’s extensions

Published online by Cambridge University Press:  01 August 2016

G. J. O. Jameson
G. J. O. Jameson*
Affiliation:
Dept of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF e-mail: g.jameson@lancaster.ac.uk
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Extract

One of the bedrock theorems of mathematics is the statement that a real polynomial of degree n has at most n real zeros. Probably the best-known proof is the algebraic one, by factorisation. But there is also a pleasant analytic proof, by deduction from Rolle’s theorem.

A slightly different question is how many positive zeros a polynomial has. Here the basic result is known as ‘Descartes′ rule of signs’. It says that the number of positive zeros is no more than the number of sign changes in the sequence of coefficients. Descartes included it in his treatise La Géométrie which appeared in 1637. It can be proved by a method based on factorisation, but, again, just as easily by deduction from Rolle’s theorem.

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Type
Articles
Information
The Mathematical Gazette , Volume 90 , Issue 518 , July 2006 , pp. 223 - 234
Copyright
Copyright © The Mathematical Association 2006

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References

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