A principal subresultant coefficient is the coefficient of 
in the 
th subresultant 
of two univariate
polynomials. For ![f,g in K[x]](/images/equations/PrincipalSubresultantCoefficient/Inline4.svg)
of degrees 
over a field 
, write
![s_j(f,g)=[x^j]S_j(f,g),](/images/equations/PrincipalSubresultantCoefficient/NumberedEquation1.svg) |
(1)
|
where ![[x^j]](/images/equations/PrincipalSubresultantCoefficient/Inline7.svg)
denotes coefficient extraction. Using the terminal
convention 
, where 
is the leading coefficient
of 
,
gives 
. In particular, 
is the resultant.
A principal subresultant coefficient can vanish even when the corresponding subresultant is nonzero, since 
may have degree less
than 
.
For example,
 |
(2)
|
so 
,
although 
.
The initial vanishing coefficients determine the degree of the polynomial greatest common
divisor. Its degree is 
iff
 |
(3)
|
where 
and the vanishing condition is empty when 
(El Kahoui 2003, pp. 290-291). Over 
, for example, the pair
 |  |  |
(4)
|
 |  |  |
(5)
|
has principal subresultant coefficients 
and subresultants 
.
Thus 
and 
.
Over an algebraic closure of 
, common roots are counted with the
minimum of their two multiplicities,
so a common multiple root can make more than one
initial coefficient vanish.
The principal subresultant coefficients are computed in the Wolfram Language using Subresultants[f, g, x].
