On the squares and cubes in the set of finite fields

Abstract

The paper provides a definition of the hinge mechanism, taking into account its kinematic nature. This definition differs significantly from that adopted by a number of mathematicians
in recent works. If we use the definition accepted today, which does not take into account the kinematic background, then the classical result of A. B. Kempe [1] about the possibility of
drawing by parts of an arbitrary plane algebraic curve with hinges of suitably chosen plane hinge mechanisms cannot be considered sufficiently substantiated by Kempe himself. This has
been noted in the modern literature [6], and even led to accusations of Kempe in error. The development and modern substantiation of Kempe’s result proposed in the works [6, 7] is, in essence, a modification of Kempe’s method for constructing the required mechanism from brick mechanisms performing algebraic actions. However, it is based on the use of a complex language of modern algebraic geometry, which leads to the replacement of Kemp’s short and transparent reasoning by an order of magnitude longer and difficult to understand texts. In our definition of the hinge mechanism, we can give a rigorous formulation of Kempe’s theorem, for the proof of which Kempe’s arguments with minimal refinements are sufficient. This updated proof is provided in the paper. The paper discusses the modern development of Kemp´e’s result, and the claims against Kemp´e’s reasoning. It also gives general ideas about mathematics that the
author has in connection with the Kemp´e theorem and its modern development.