Generalizations of Rolle's theorem  

AMS classification numbers  Link : 

12D10  Link , 12J15  Link , 26C10  Link , 
30C10  Link , 30C15  Link , 46G05  Link .

Page maintained by Jean-Claude Evard. Last update: June 6, 2003.

  Rolle's theorem in finite dimensional spaces 

 1995 
Massimo Furi and Mario Martelli, 
A multidimensional version of Rolle's theorem,
Amer. Math. Monthly 102 (3): 243--249, 1995.

 Algebraic versions of Rolle's theorem 

 1995 
Paulo Ribenboim, 
On Rolle fields, with examples,
Arch. Math. 64 (5): 402--409, 1995.

 1990 
Danielle Gondard-Cozette, 
Axiomatisations simples des théories des corps de Rolle, 
(French. English summary) [Simple axiomatizations of Rolle field theories], 
Manuscripta Math. 69 (3): 267--274, 1990.

 1984 
Ron Brown, Thomas C. Craven, and M. J. Pelling, 
Ordered fields satisfying Rolle's theorem, 
Ordered fields and real algebraic geometry, 
Rocky Mountain J. Math. 14 (4): 819--820, 1984.

  Complex versions of Rolle's theorem 

Grace-Heawood Theorem: Let p be a polynomial with complex coefficients and
degree n greater than one. Let z_1 and z_2 be two distinct complex numbers such 
that p(z_1) = p(z_2). Let c denote the midpoint of the line segment from z_1 to z_2.
Let d denote the distance between z_1 and z_2. Let r = (d/2)cot(pi/2). Let D denote
the closed disk with center c and radius r in the complex plane. Then p has at least
one critical point in D.

A proof of Grace-Heawood Theorem can be found in [M] and [RS].

Comments (Summarized from [RS] page 138): Grace-Heawood theorem was 
first published by J. H. Grace in 1902. P. J. Heawood found it and published it 
independently, with a different proof, in 1907.

 Books 

 [M] Morris Marden,
       Geometry of polynomials,
       Second edition,
       Mathematical Survey,

[RS]  Q. I. Rahman and Gerhard Schmeisser, 
          Analytic theory of polynomials, 
          The Clarendon Press, Oxford University Press, Oxford, 2002. xiv+742 pp.

 Papers 

 1997 
D. Novikov and S. Yakovenko, 
A complex analogue of the Rolle theorem and polynomial envelopes 
of irreducible differential equations in the complex domain,
J. London Math. Soc. 56 (2): 305--319, 1997.

 1985 
Morris Marden, 
The search for a Rolle's theorem in the complex domain, 
Amer. Math. Monthly 92 (9): 643--650, 1985.

 1907 
P. J. Heawood,
Geometrical relations between the roots of f(x) = 0, f ' (x) = 0,
Quarterly Journal of Mathematics, 38: 84--107, 1907.

 1902 
J. H. Grace,
The zeroes of a polynomials,
Proceedings of the Cambridge Philosophical Society, 11: 352--357, 1902.

 The failure of Rolle's theorem in infinite-dimensional spaces 

 2002 
Jesús Ferrer, 
Rolle's theorem for polynomials of degree four in a Hilbert space,
J. Math. Anal. Appl. 265 (2002), no. 2, 322--331.

 2001 
Daniel Azagra and Mar Jimenez-Sevilla
The failure of Rolle's theorem in infinite-dimensional Banach spaces
Journal of Functional Analysis, 182: 207-226, 2001.
pdf-copy  Link  posted on the Web site of Mar Jiménez-Sevilla  Link .

Franck Wielonsky, 
A Rolle's theorem for real exponential polynomials in the complex domain,  
J. Math. Pures Appl. (9) 80 (4): 389--408, 2001.

 2000 
Jesús Ferrer, 
On Rolle's theorem in spaces of infinite dimension,
B. N. Prasad birth centenary commemoration volume II, 
Indian J. Math. 42 (1): 21--36, 2000.

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