The Bolzano-Poincaré Type Theorems

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.