# The resultant

For univariate polynomials over a field the following lemma says that it is possible to find polynomials such that , , if and only if .

Lemma 1   Let be nonzero polynomials over the field k. Then the following statements are equivalent.
• There exist polynomials such that

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Proof. Let . If then and a solution is

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Conversely, let be a solution. Assume that and are coprime. Then would imply . This is impossible since and . Hence and are not coprime and holds.

Remark 7   Given nonzero polynomials of degrees respectively we consider the map

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For we define

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with the convention that

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The restriction of to is a linear map between vector spaces of finite dimension.

Proposition 4   Let nonzero polynomials of degrees such that . Then we have:
• is an isomorphism.
• If then the Bézout coefficients computed by the Extended Euclidean Algorithm form the unique solution in of the equation

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Proof. From Lemma 1 we deduce that the kernel of is reduced to iff . Since both and have dimension this proves the first claim. The second claim is a consequence of the first one.

Remark 8   Let's carry on with nonzero univariate polynomials in of degrees such that . However let us relax the hypothesis on the coefficient ring by assuming that it is just a commutative ring with identity element. Let us write:

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The natural basis for consists of the for followed by the for . On this basis is represented by the following matrix

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where all entries outside of the parallelograms are equal to zero.

Definition 4   The above square matrix of order is denoted and called the Sylvester matrix of and . Its determinant is called the resultant of and denoted by . We make the following conventions.
• If (and still nonzero polynomials) we define .
• If or then .
• If then .

Proposition 5   Let be nonzero univariate polynomials over a field . Then the following statements are equivalent
1. .
2. .
3. there do not exist any such that

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Proof. This follows from Proposition 4 and the fact that is a determinant of the linear map .

Proposition 6   Let be nonzero univariate polynomials over a UFD . Then we have

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Proof. This follows from the adaptation of the results of Lemma 1 and Proposition 4 to the case where the ground ring is a UFD (and in particular an integral domain) instead of a field.

Proposition 7   Let be nonzero univariate polynomials over an integral domain . Then there exist nonzero such that

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Proof. Let be the field of fractions of . If then we know that there exist such that

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Then the claim follows by cleaning up the denominators.

If then and are coprime in . Then there exists with stated degree bounds such that holds in . Observe that

• The coefficients of are in fact the unique solution of a linear system whose matrix is the Sylvester matrix of and .
• These coefficients can be computed by the Cramer's rule.
• Hence each of these coefficients is the quotient of a determinant of a submatrix of by .
Then the polynomials and have coefficients in and we have the desired relations.

Marc Moreno Maza
2008-01-07