We now consider two independent sets of indeterminates
and 
.
One defines for each 
the double Schubert polynomial 
to
be
where 
denotes the longest element of 
, divided
differences act only on the x variables and
These polynomials which belong to the polynomial ring
satisfy the following properties:
, 
is a non-zero homogeneous polynomial in
of
degree 
,
's to 0, 
since
,
and 
,
, 
.
Double Schubert polynomials generalize the coefficients
in the Newton interpolation formula (1.10)
to the case of any number of variables. In other words,
considering one set of indeterminates 
and
denoting any set of elements of a field 
,
the Newton interpolation formula (1.10) becomes
for 
the equation
in which divided differences act on the y variables.
Thus, working with two sets of variables involves only adding an extra term in Monk's formula (1.38). Furthermore, as in the case of simple Schubert polynomials, one may choose r such that
In the same manner as for the simple Schubert polynomials, it gives a way to
efficiently compute the expression of a double Schubert polynomial in the
basis of monomials but the most interesting point to notice is that it also provides an
efficient way to specialize
(Specialize )
double Schubert polynomials, since in the relation
(1.45) the term 
will be very soon specialized to
zero whenever the specialization of the second alphabet to a permutation of the
first one sends the factor 
onto zero.
The TableXX function gives the table
of all double Schubert polynomials for 
.
