We know that the resultant of two polynomials can be computed as the determinant of their Sylvester matrix ( <http://en.wikipedia.org/wiki/Sylvester_matrix> ). How do we compute the resultant of more than two polynomials? Can we generalise the use of the Sylvester matrix for this case?



Multipolynomial resultants
To speak of a (single) resultant of several polynomials one must allow for several variables as well and the situation becomes trickier that in the one variable case.


Let $f=(f\_0,\ldots, f\_k)$ be homogeneous polynomials (say over an algebraically closed field) of degrees $d\_0,\ldots, d\_k$ in $k+1$ variables. One can view $f$ as a section of $E=\mathcal{O(d\_0)}\oplus\cdots\oplus\mathcal{O(d\_k)}$, a vector bundle on $\mathbf{P}^k$. Starting from this bundle form the positive Koszul complex $$K=[0\to \mathcal{O}\_{\mathbf{P}^k}\to E\to \wedge^2 E\to\cdots\to\wedge^{k+1}E\to 0]$$


where $\mathcal{O}\_{\mathbf{P}^k}$ is in degree 0 and the differential $\wedge^iE\to\wedge^{i+1}E$ is given by the wedge with $f$. This complex is (stalkwise) acyclic iff $f$ is everywhere non-zero see e.g. Proposition 1.4 of Gelfand, Kapranov, Zelevinsky, Discriminants, resultants and multidimensional determinants, chapter 2. If $K$ is acyclic, then its hypercohomology vanishes; on the other hand the higher cohomology of each of the sheaves in $K$ vanishes (see e.g. Hartshorne, chapter 3). So if $K$ is acyclic, then the complex $\Gamma K$of the global sections of $K$ is acyclic. On the other hand, since all sheaves in $K$ are globally generated, the converse is also true: if $\Gamma K$ is acyclic, then $K$ is acyclic.


The complex $\Gamma K$ can be described explicitly as follows: the degree $i$ part is $\bigoplus\_{|A|=i}C\_{A}$ where $A\subset{ \{ } 0,\ldots, k\}$, $C\_{A}$ is the space of all homogeneous polynomials of degree $\sum\_{j\in A} d\_{j}$ and the differential restricted to $ C\_A$ is the sum for all $j\not\in A$ of the maps $ C\_A\to C\_{A\cup\{j\}}$ given by multiplication by $f\_j$ times $(-1)^{b}$ where $b$ is the number of the elements of $A$ that are greater than $j$.


The value of the resultant on $f$ is the hyperdeterminant of $\Gamma K$. (Hyperdeterminants are a generalization of the usual determinants to complexes; they are described e.g. in Gelfand, Kapranov, Zelevinsky, Appendix A.)


