linguistic:paraphrasing
Resultant of generic homogeneous polynomials

A homogeneous polynomial of degree d in n variables may have up to (n + d − 1 n − 1) = (n + d − 1) ! (n − 1) ! d ! {\displaystyle {\binom {n+d-1}{n-1}}={\frac {(n+d-1)!}{(n-1)!\,d!}}} coefficients; it is said to be generic, if these coefficients are distinct indeterminates.

Let P 1, …, P n {\displaystyle P_{1},\ldots,P_{n}} be n generic homogeneous polynomials in n indeterminates, of respective degrees d 1, …, d n. {\displaystyle d_{1},\dots,d_{n}.} Together, they involve ∑ i = 1 n (n + d i − 1 n − 1) {\displaystyle \sum _{i=1}^{n}{\binom {n+d_{i}-1}{n-1}}} indeterminate coefficients.Let C be the polynomial ring over the integers, in all these indeterminate coefficients. The polynomials P 1, …, P n {\displaystyle P_{1},\ldots,P_{n}} belong thus to C, {\displaystyle C,} and their resultant (still to be defined) belongs to C.

The Macaulay degree is the integer D = d 1 + ⋯ + d n − n + 1, {\displaystyle D=d_{1}+\cdots +d_{n}-n+1,} which is fundamental in Macaulay's theory. For defining the resultant, one considers the Macaulay matrix, which is the matrix over the monomial basis of the C -linear map (Q 1, …, Q n) ↦ Q 1 P 1 + ⋯ + Q n P n, {\displaystyle (Q_{1},\ldots,Q_{n})\mapsto Q_{1}P_{1}+\cdots +Q_{n}P_{n},} in which each Q i {\displaystyle Q_{i}} runs over the homogeneous polynomials of degree D − d i, {\displaystyle D-d_{i},} and the codomain is the C -module of the homogeneous polynomials of degree D.

If n = 2, the Macaulay matrix is the Sylvester matrix, and is a square matrix, but this is no longer true for n > 2. Thus, instead of considering the determinant, one considers all the maximal minors, that is the determinants of the square submatrices that have as many rows as the Macaulay matrix. Macaulay proved that the C -ideal generated by these principal minors is a principal ideal, which is generated by the greatest common divisor of these minors. As one is working with polynomials with integer coefficients, this greatest common divisor is defined up to its sign. The generic Macaulay resultant is the greatest common divisor which becomes 1, when, for each i, zero is substituted for all coefficients of P i, {\displaystyle P_{i},} except the coefficient of x i d i, {\displaystyle x_{i}^{d_{i}},} for which one is substituted.