Open AccessThe Bolzano-Poincaré Type Theorems
Author(s) -
Przemysław Tkacz,
Marian Turzański
Publication year - 2011
Publication title -
international journal of mathematics and mathematical sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.21
H-Index - 39
eISSN - 1687-0425
pISSN - 0161-1712
DOI - 10.1155/2011/793848
Subject(s) - mathematics , equivalence (formal languages) , poincaré conjecture , brouwer fixed point theorem , type (biology) , fundamental theorem , pure mathematics , polynomial function theorems for zeros , fixed point theorem , discrete mathematics , mathematical analysis , ecology , biology , alternating polynomial , matrix polynomial , polynomial
In 1883–1884, Henri Poincaré announced the result about the structure of the set of zeros of function ∶→, or alternatively the existence of solutions of the equation ()=0. In the case =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 =3 an algorithmical proof of the Bolzano-Poincaré Theorem and to show the equivalence of Poincaré, Brouwer and “Onthe existence of a chain” theorems
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