The Bolzano-Poincaré Type Theorems

Przemysław Tkacz and Marian Turzański

International Journal of Mathematics and Mathematical Sciences, 2011, vol. 2011, 1-9

Abstract:

In 1883–1884, Henri Poincaré announced the result about the structure of the set of zeros of function ð ‘“ ∶ ð ¼ ð ‘› → ð ‘… ð ‘› , or alternatively the existence of solutions of the equation ð ‘“ ( ð ‘¥ ) = 0 . In the case ð ‘› = 1 the Poincaré Theorem is well known Bolzano Theorem. In 1940 Miranda rediscovered the Poincaré Theorem. Except for few isolated results it is essentially a non-algorithmic theory. The aim of this article is to introduce an algorithmical proof of the Theorem “On the existence of a chain” and for ð ‘› = 3 an algorithmical proof of the Bolzano-Poincaré Theorem and to show the equivalence of Poincaré, Brouwer and “On the existence of a chain” theorems.

Date: 2011
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:793848

DOI: 10.1155/2011/793848

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