nLab resultant

Idea

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

• wikipedia