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We provide, in a 474 pages study, a comprehensive and self-containedtreatment of Resultant Theory for a homogeneous system of polynomials withseveral variables (as many variables as of polynomials).  In a non classical way, we use the multiplicative structure of finite freeresolutions, by applying it to the complex homogeneous components of the Koszulcomplex of the system, and this in any degree. Moreover, these complexes haveMacaulay decompositions. These three pillars, multiplicative structure, Koszulcomplex, Macaulay decomposition, allow to establish, surprisingly to us,remarkable binomial relations between 3 families of scalars resulting from thedifferentials of that complexes. These binomial relations generate severalnotable results, like a determinantal expression of a certain denominator,depending only on the first differential, and provide in particular formulasexpressing the resultant. We have explored more deeply the case of the criticaldegree delta. This degree produces a fundamental linear form on the homogeneouspolynomial component of degree delta, including the resultant. And thisfundamental linear form has allowed us to highlight a certain number ofremarkable properties.  Key words: Elimination theory, homogeneous polynomial system, resultant,determinant of complexes, Cayley determinant, computational algebra, finitefreeresolution, Koszul complex, Euler characteristic, Grade, Multiplicativestructure, Macaulay decomposition, Fitting invariants, MacRae invariant,regular sequence.

Une approche combinatoire pour le r\'esultant multivari\'e. Le r\'esultant multivari\'e pour les enfants motiv\'es