Summary:
Abstract
This thesis mainly refers to two issues related to solving algebraic geometric
problems.
In the first, which is related to the algorithmic proof of theorems of the
Euclidean geometry, we present one of the most important tools of computational
algebraic geometry, the algorithm Buchberger. In this algorithm the input is a
system of multivariate polynomial equations and the output an equivalent system
of equations called Groebner basis. The Buchberger algorithm is a nice
generalization of the algorithms of finding the greatest common divisor in the
case of polynomials of a variable and the Gauss elimination for solving systems
of linear polynomial equations with many unknowns.
With regards to the second issue, we exhibit a simple version of Hilbert's
third problem. Actually, we consider whether it is possible to find a formula
to calculate the volume of any polyhedron, by applying elementary methods,
without resorting to infinite processes. By citing a proof registered to Dehn
Max, we eventually prove that this is impossible since, with these methods, we
can not "transform" a regular tetrahedron into a cube with the same volume.
Keywords:
Hilbert third problem, algorium Buchberger , Groebner basis, formalism, platonism
