The resultant of two monic polynomials is an expression which is zero if the polynomials have a common multiple: it may be defined as a product of all possible difference between the pairs of roots of the two polynomials. They are an analogue of a discriminant of a single quadratic equation: the discriminant is zero if the equation has a double root. Resultant for $n$ homogeneous polynomials of $n$ variables defines a condition of solvability for a system of $n$ homogeneous polynomials, as the determinant does for a system of linear equations.

When the number of polynomials and variables is bigger than few smallest integers, it is very little known. There are remarkable determinant expressions for resultants of 2 or 3 polynomials. In what is the first article on homological algebra, Cayley has shown how to descibe a resultant as a determinant of a complex which is a certain Koszul resolution.

While syzygies show linear relations, linear relations among linear relations and so on and thus belong to linear homological algebra, resultants seem to point to a nonlinear homological algebra for polynomial relations among polynomials and so on.

References

I. M. Gelfand, M. M. Kapranov, A. Zelevinsky, Discriminants, resultants, and multidimensional determinants, Birkhäuser 1994, 523 pp.

A. Anokhina, A. Morozov, Sh. Shakirov, Resultant as determinant of Koszul complex, ITEP/TH-70/08. Theor.Math.Phys. 160:3 (2009) 1203-1228, arXiv:0812.5013

A.Morozov, Sh.Shakirov, New and old results in resultant theory, arxiv/0911.5278

H. B. Griffiths, Cayley’s version of the resultant of two polynomials, Amer. Math. Monthly 88, No. 5 (May, 1981), pp. 328-338, jstor

A. Morozov, Sh. Shakirov, Resultants as contour integrals, arXiv:0807.4539