In this post, we shall see how the resultant, a determinant containing coefficients of two polynomials is useful in determining the existence of their common roots.

Suppose we are working in a ring and are two polynomials in :

Consider the system of equations obtained by multiplying the first equation by and the second by . Make the following change of variables:

The equations: can be denoted in matrix form as:

Call the above matrix as . If there was a common solution so that , then above system of equations would have a non-zero solution (non-zero since ), so the determinant of would have to be zero. If the ring is an integral domain, then the vanishing of the determinant of is equivalent to and having a common root.

This determinant is known as the resultant of and . It has some interesting consequences that I state below:

**Theorem 1** For two non-constant irreducible polynomials to have a common root, it is necessary and sufficient that their resultant vanishes.

Assume that are irreducible and consider where . Thus the coefficients of and are polynomials in . The matrix similar to above has coefficients polynomials in , so here the resultant is a polynomial in , say . If is a common root of and , then . Hence the -coordinate of any common root must be a root of . In particular, there can be at most finitely many ‘s so that is a common root.

Further, for every value of so that , there are only finitely many values of so that is a root of . Hence and can intersect in finitely many points! We record this as one version of Bezout’s theorem:

**Theorem 2 (Bezout)** If and are irreducible polynomials of degrees in , then they can intersect in at most points.

Here, we have proved only finiteness of common points. For the refinement , see Abhyankar‘s book *Lectures in Algebra*.

**Theorem 3** Any curve can have at most finitely many singular points.

Here, we define a point to be singular if it satisfies . The proof follows from Bezout’s theorem when we set or .

Resultants can also be used to compute the discriminant of a polynomial. The discriminant of a polynomial

is given by

Upto a sign factor , we have

The Wikipedia article on Resultants takes the Theorem 1 to be the definition and states different results.

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