The resultant

For univariate polynomials f, g over a field the following lemma says that it is possible to find polynomials s, t such that sf + tg = 0, deg(s) < deg(g), deg(t) < deg(f ) if and only if gcd(f, g) $\neq$ 1.

Lemma 1   Let f, g $\in$ $\bf k$[x] be nonzero polynomials over the field k. Then the following statements are equivalent.

• gcd(f, g) $\neq$ 1
• There exist polynomials s, t $\in$ R[x] $\setminus$ {0} such that

  sf + tg  =  0,    deg(s) < deg(g)    and    deg(t) < deg(f ) (24)

  
  

Proof. Let h = gcd(f, g). If h $\neq$ 1 then deg(h) $\geq$ 1 and a solution is

(s, t)  =  (g/h, - f /h) (25)

Conversely, let (s, t) be a solution. Assume that f and g are coprime. Then sf = - tg would imply f | t. This is impossible since t $\neq$ 0 and deg(t) < deg(f ). Hence f and g are not coprime and gcd(f, g) $\neq$ 1 holds. $\qedsymbol$

Remark 7   Given f, g $\in$ $\bf k$[x] nonzero polynomials of degrees n, m respectively we consider the map

$$\phi \left.\vphantom{ \begin{array}{lll} {\bf k}[x] \times {\bf k}[x] ... ...tarrow & {\bf k}[x] \\ (s,t) & \longmapsto & s f + tg \\ \end{array} }\right. \begin{array}{lll} {\bf k}[x] \times {\bf k}[x] & \rightarrow & {\bf k}[x] \\ (s,t) & \longmapsto & s f + tg \\ \end{array}$$ (26)

For d $\in$ $\mbox{${\mathbb N}$}$ $\setminus$ {0} we define

$$\in \bf k$$ (27)

with the convention that

P₀  =  {0} (28)

The restriction of $\phi$ to  P_m×P_n  $\rightarrow$  P_n+m is a linear map $\phi_{{0}}^{{}}$ between vector spaces of finite dimension.

Proposition 4   Let f, g $\in$ $\bf k$[x] nonzero polynomials of degrees n, m such that n + m > 0. Then we have:

• gcd(f, g) = 1  $\iff$  $\phi_{{0}}^{{}}$ is an isomorphism.
• If gcd(f, g) = 1 then the Bézout coefficients (u, v) computed by the Extended Euclidean Algorithm form the unique solution in P_m×P_n of the equation

  $$\phi_{{0}}^{{}}$$ (29)

  
  

Proof. From Lemma 1 we deduce that the kernel of $\phi_{{0}}^{{}}$ is reduced to {0} iff gcd(f, g) = 1. Since both P_m×P_n and P_n+m have dimension n + m this proves the first claim. The second claim is a consequence of the first one. $\qedsymbol$

Remark 8   Let's carry on with f, g nonzero univariate polynomials in x of degrees n, m such that n + m > 0. However let us relax the hypothesis on the coefficient ring by assuming that it is just a commutative ring R with identity element. Let us write:

f  =  f_nxⁿ + ^... + f₀    and    g  =  g_mx^m + ^... + g₀ (30)

The natural basis for P_m×P_n consists of the (xⁱ, 0) for 0 $\leq$ i < m followed by the (0, x^j) for 0 $\leq$ j < m. On this basis $\phi_{{0}}^{{}}$ is represented by the following matrix

$$\left(\vphantom{ \begin{array}{cccccccc} f_0 & 0 &\cdots & 0 & g_... ... \ddots & \\ 0 &\cdots & 0 & f_n & 0 &\cdots &0 & g_m \\ \end{array} }\right. \begin{array}{cccccccc} f_0 & 0 &\cdots & 0 & g_0 & 0 &\cdots & 0... ...&\ddots & \ddots & \\ 0 &\cdots & 0 & f_n & 0 &\cdots &0 & g_m \\ \end{array} \left.\vphantom{ \begin{array}{cccccccc} f_0 & 0 &\cdots & 0 & g_... ... \ddots & \\ 0 &\cdots & 0 & f_n & 0 &\cdots &0 & g_m \\ \end{array} }\right)$$ (31)

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

Definition 4   The above square matrix of order n + m is denoted Sylv(f, g) and called the Sylvester matrix of f and g. Its determinant is called the resultant of f and g denoted by res(f, g). We make the following conventions.

• If n = m = 0 (and still nonzero polynomials) we define res(f, g) = 1.
• If f = 0 or f $\in$ R[x] $\setminus$ R then res(f, 0) = res(0, f )= 0.
• If f $\in$ R $\setminus$ {0} then res(f, 0) = res(0, f )= 1.

Proposition 5   Let f, g $\in$ $\bf k$[x] be nonzero univariate polynomials over a field $\bf k$. Then the following statements are equivalent

1. gcd(f, g) = 1.
2. res(f, g) $\neq$ 0.
3. there do not exist any s, t $\in$ $\bf k$[x] $\setminus$ {0} such that

   sf + tg = 0,  deg(s) < deg(g)    and    deg(t) < deg(f ). (32)

   
   

Proof. This follows from Proposition 4 and the fact that res(f, g) is a determinant of the linear map $\phi_{{0}}^{{}}$. $\qedsymbol$

Proposition 6   Let f, g $\in$ R[x] be nonzero univariate polynomials over a UFD R. Then we have

$$\in \iff \neq$$ (33)

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. $\qedsymbol$

Proposition 7   Let f, g $\in$ R[x] be nonzero univariate polynomials over an integral domain R. Then there exist nonzero s, t $\in$ R[x] such that

sf + tg = res(f, g),  deg(s) < deg(g)    and    deg(t) < deg(f ). (34)

Proof. Let $\bf k$ be the field of fractions of R. If res(f, g) = 0 then we know that there exist s, t $\in$ $\bf k$[x] $\setminus$ {0} such that

sf + tg = 0,  deg(s) < deg(g)    and    deg(t) < deg(f ). (35)

Then the claim follows by cleaning up the denominators.

If res(f, g) $\neq$ 0 then f and g are coprime in $\bf k$[x]. Then there exists u, v $\in$ $\bf k$[x] with stated degree bounds such that uf + vg = 1 holds in $\bf k$[x]. Observe that

• The coefficients of u, v are in fact the unique solution of a linear system whose matrix is the Sylvester Sylv(f, g) matrix of f and g.
• 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 Sylv(f, g) by res(f, g).

Then the polynomials s = res(f, g) u and t = res(f, g) v have coefficients in R and we have the desired relations. $\qedsymbol$