stylistic:tone, structural:reformatting
Elimination property

Let I = ⟨ A, B ⟩ {\displaystyle I=\langle A,B\rangle } be the ideal generated by two polynomials A and B in a polynomial ring R, {\displaystyle R,} where R = k {\displaystyle R=k} is itself a polynomial ring over a field. If at least one of A and B is monic in x, then:

The first assertion is a basic property of the resultant. The other assertions are immediate corollaries of the second one, which can be proved as follows.

As at least one of A and B is monic, a tuple (β 1, …, β n) {\displaystyle (\beta _{1},\ldots,\beta _{n})} is a zero of res x (A, B) {\displaystyle \operatorname {res} _{x}(A,B)} if and only if there exists α {\displaystyle \alpha } such that (β 1, …, β n, α) {\displaystyle (\beta _{1},\ldots,\beta _{n},\alpha)} is a common zero of A and B. Such a common zero is also a zero of all elements of I ∩ R. {\displaystyle I\cap R.} Conversely, if (β 1, …, β n) {\displaystyle (\beta _{1},\ldots,\beta _{n})} is a common zero of the elements of I ∩ R, {\displaystyle I\cap R,} it is a zero of the resultant, and there exists α {\displaystyle \alpha } such that (β 1, …, β n, α) {\displaystyle (\beta _{1},\ldots,\beta _{n},\alpha)} is a common zero of A and B. So I ∩ R {\displaystyle I\cap R} and R res x (A, B) {\displaystyle R\operatorname {res} _{x}(A,B)} have exactly the same zeros.