For two polynomials, what is the relationship between the least linear combination and the resultants? Let $k$ be a field and $k[x]$ be the ring of polynomials over $k$. Given two polynomis $m_1(x), m_2(x) \in k[x]$,  I want to know the relationship between the resultants and the least linear combination.
(In Euclidean rings, the least linear combination is the greatest common divisor (GCD).)
 A: The important notion here is that of subresultant. Suppose $P_1, P_2$ are two polynomials of degrees $d_1,d_2$ and suppose $d_1\ge d_2$. To compute the GCD you would typically use Euclid's division algorithm: you divide $P_1$ by $P_2$ and get a remainder $P_3$ then you divide $P_2$ by $P_3$ and get the remainder $P_4$ etc. The last nonzero remainder is the GCD. Now imagine doing that for generic polynomials
$$
P_1(x)=a_{1,d_1}x^{d_1}+a_{1,d_1-1}x^{d_1-1}+\cdots+a_{1,1}x+a_{1,0}
$$
and
$$
P_2(x)=a_{2,d_2}x^{d_2}+a_{2,d_2-1}x^{d_2-1}+\cdots+a_{2,1}x+a_{2,0}\ .
$$
The very first step would be to subtract $\frac{a_{1,d_1}}{a_{2,d_2}}x^{d_1-d_2}P_2(x)$ from $P_1(x)$. Don't do that. Instead multiply $P_1$ by $a_{2,d_2}$ and then subtract $a_{1,d_1}x^{d_1-d_2}P_2(x)$ so as not to produce fractions. Rince and repeat.
Generically the degree of the remainder only drops by one. The resultant essentially is the degree zero remainder, i.e., $P_{d_2+2}$. The previous $P$'s are the subresultants (up to one's choice of normalization convention, there may also be extraneous factors to peel off).
A good reference on the subject is the book "Algorithms in Real Algebraic Geometry" by Basu, Pollack and Roy.
