The Rouché-Fontené theorem plays a fundamental role in the study of linear equations. It gives, in fact, the general rule for solving linear systems of the type n×p (that is to say, systems of linear equations in n unknowns p, (n,p)∈N^*×N^*). This rule is based on the determination and calculation of the respective values of critical characteristics or "trimmed" and, of course, on their list, given that, as they are all zero or one of them do not be the result of a linear system which they are extracted radically changes. The demonstration to prove that the set μ(p,q,r) real matrices p×q type and rank r is a variety diving is based on the same type of reasoning based on the identification and calculation tacking matrices determinants; It can therefore rightly be regarded as a unique illustration and / or application of an aspect of Rouché-Fontené theorem. The aim of this article is precisely to show how Rouché-Fontené theorem (mainly within linear algebra) is a scope in the dives varieties theory (rather under differential geometry).

The mathematical notion of "dual" and "duality" or "proposal dual" in its federating release, unifying, generalizing and simplifying (fugs, in acronym) studied in Linear Algebra (specifically in the areas of theory vector), is a concept that is a common denominator in several branches of mathematics. It appears, in fact, under diverse forms and with apparent differences on the semantic level, both in projective geometry, set theory, topology in differential geometry and crystallography, to name but a few illustrative examples. It is legitimate to ask about the relationship between these different aspects of duality and our purpose in this article is precisely to show that the duality which we call algebraic - that is to say one that is studied in universities in the theory of vector spaces - is one to which identify themselves in one way or another, all other forms of duality studied mathematics. While it must be acknowledged here that, in some cases, the relationship between some of these forms of dualities with algebraic duality is not always easy to establish at first.