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