The Resultant via a Koszul Complex

Abstract

As noticed by Jouanolou, Hurwitz proved in 1913 ([Hu]) that, in the generic case, the Koszul complex is acyclic in positive degrees if the number of (homogeneous) polynomials is less than or equal to the number of variables^†. It was known around 1930 that resultants may be calculated as a Mc Rae invariant of this complex. This expresses the resultant as an alternate product of determinants coming from the differentials of this complex. Demazure explained in a preprint ([De]), how to recover this formula from an easy particular case of deep results of Buchsbaum and Eisenbud on finite free resolutions. He noticed that one only needs to add one new variable in order to do the calculation in a non generic situation.

I have never seen any mention of this technique of calculation in recent reports on the subject (except the quite confidential one of Demazure and in an extensive work of Jouanolou, however from a rather different point of view). So, I will give here elementary and short proofs of the theorems needed—except the well-known acyclicity of the Koszul complex and the “Principal Theorem of Elimination”—and present some useful remarks leading to the subsequent algorithm. In fact, no genericity is needed (it is not the case for all the other techniques). Furthermore, when the resultant vanishes, some information can be given about the dimension of the associated variety.

As an illustration of this technique, we give an arithmetical consequence on the resultant: if the polynomials have integral coefficients and their reductions modulo a prime p defines a variety of projective dimension zero and degree d, then p ^d divides the resultant.