Waring's Theorem (Analysis)

The Waring set is a mathematical theorem from the mathematical sub-area of the analysis corresponding to the mathematician Edward Waring is allocated. The sentence is closely related to the sentence of Rolle .

Formulation of the sentence

The sentence can be stated as follows:

Let a real polynomial function and three real numbers be given . ${\displaystyle f\colon \mathbb {R} \to \mathbb {R} }$ ${\displaystyle a,b,c\in \mathbb {R} }$

The following should apply:

(I) . ${\displaystyle a<b}$

(II) and are zeros of . ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle f}$

(III) There is no zero of in the open interval . ${\displaystyle ]a,b[}$ ${\displaystyle f}$

(IV) is the associated polynomial function with . ${\displaystyle g\colon \mathbb {R} \to \mathbb {R} }$ ${\displaystyle g(x)=f'(x)+c\cdot f(x)\;(x\in \mathbb {R} )}$

Then:

${\displaystyle g}$has an odd number of zeros in the open interval and - especially! - always at least one. ${\displaystyle ]a,b[}$

literature

• Siegfried Gottwald , Hans-Joachim Ilgauds and Karl-Heinz Schlote (ed.): Lexicon of important mathematicians . Verlag Harri Deutsch , Thun 1990, ISBN 3-8171-1164-9 , p. 482 ( MR1089881 ).
• Franz Xaver Mayer : Eduard Warings Meditationes algebraicae . Inaugural dissertation ( University of Zurich ). Überlingen (Lake Constance) 1923 ( WorldCat-Link ).

Remarks

1. ↑ The article in the lexicon of important mathematicians (p. 482) draws on the above-mentioned dissertation by Franz Xaver Mayer.
2. ↑ Here - as usual - stands for the associated derivative function , which is also a real polynomial function. ${\displaystyle f'}$ ${\displaystyle f}$

Individual evidence

1. ↑ ^{a b} Siegfried Gottwald et al. (Ed.): Lexicon of important mathematicians. 1990, p. 482