EXPLICIT FORMULAS FOR THE MULTIVARIATE RESULTANT

CARLOS D’ ANDREA AND ALICIA DICKENSTEIN

Abstract

Given n homogeneous polynomials f_{1}, …, f_{n} in n variables over an algebraically closed field k with respective degrees d_{1}, …, d_{n}, the resultant Res_{d_{1},...,d_{n}}(f_{1}, …, f_{n}) is an irreducible polynomial in the coefficients of f_{1}, …, f_{n}, which vanishes whenever f_{1}, …, f_{n} have a common root in projective space. Classical Macaulay formulas describe the resultant as an explicit quotient of two determinants. These formulas involve a matrix of size at least the number of monomials in n variables of degree t_{n} + 1, and a submatrix of it. Here, t_{n} denotes the critical degree ∑_{i}(d_{i} − 1). The non-zero entries in these matrices are coefficients of the homogeneous polynomials. Macaulay’s work has been revisited and sharpened by Jouanolou, who proposed for each t ≥ 0, a square matrix M_{t} whose determinant is a nontrivial multiple of Res_{d_{1},...,d_{n}}(f_{1}, …, f_{n}). In this case, the non-zero entries are coefficients of the given polynomials and of their Bezoutian.

We explicitly find the extraneous factor in Jouanolou’s formulation, i.e. the polynomial det(M_{t})/Res_{d_{1},...,d_{n}}(f_{1}, …, f_{n}), which again happens to be the determinant of a submatrix of M_{t} for every t, and we thus give new formulas à la Macaulay for the resultant. For t > t_{n}, we recover Macaulay’s classical formulas. For t ≤ t_{n}, we prove that the size of the matrix M_{t} is considerably smaller.

We identify these matrices as square submatrices of one the morphisms in a Koszul resultant complex which in general has many non zero terms, and whose determinant is Res_{d_{1},...,d_{n}}(f_{1}, …, f_{n}). These are the complexes considered by Weyman and Zelevinsky in the equal degree case. This allows us to classify all data d_{1}, …, d_{n}, t for which Res_{d_{1},...,d_{n}}(f_{1}, …, f_{n}) = det(M_{t}).

We show how different classical formulas can be viewed as special cases of the determinantal formulas that we present. In particular, we also recover in this setting the “affine” Dixon formulas.