A subresultant is a polynomial generalizing the resultant of two univariate polynomials. Subresultants provide a way to compute their polynomial greatest common divisor using determinants formed from their coefficients (Collins 1967, Brown and Traub 1971).
Let ![f,g in K[x]](/images/equations/Subresultant/Inline1.svg)
have polynomial
degrees 
,
where 
is a field. For 
, form the matrix 
whose rows are the coefficients
of
 |
(1)
|
with columns in descending powers from 
to 1. This matrix has
rows and 
columns. Let 
consist of its first 
columns followed by its column for 
. The 
th subresultant is
 |
(2)
|
It has degree at most 
, and each coefficient is a
polynomial in the coefficients
of 
and 
(El Kahoui 2003, pp. 283-285). In particular, 
is the Sylvester matrix,
so 
is the resultant.
With 
the leading coefficient of 
, the terminal convention
 |
(3)
|
gives a sequence with 
members. When 
, this last member is 
. Other normalizations of the subresultant sequence
are also used.
The principal subresultant coefficient 
is the coefficient
of 
in 
. Writing 
, some simple examples are
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
These are lists of principal subresultant coefficients, rather than lists of the subresultant polynomials themselves. For example, the subresultant polynomial at index 1 for the two quadratic polynomials is
 |
(7)
|
If 
is the degree
of the polynomial greatest common
divisor, then all subresultants of index less than 
vanish, and 
is a nonzero scalar multiple of that divisor (El Kahoui
2003, pp. 290-291). Equivalently, the first nonzero principal
subresultant coefficient has index 
. Over an algebraic closure
of 
, this counts common roots
with each multiplicity equal to the smaller of its
multiplicities in 
and 
.
The subresultant polynomials are computed in the Wolfram Language using SubresultantPolynomials[f, g, x], while their principal subresultant coefficients are computed using Subresultants[f, g, x].
