pseudoRemainder  compute the pseudoremainder
Synopsis

 Usage:
pseudoRemainder(f,g)

Inputs:

Outputs:

a ring element, the pseudo remainder of the polynomial f by the polynomial g
Description
Let
x be the first variable of
R appearing in
g. Suppose that
g has degree
d in
x, and that the coefficient of
x^d in
g (as an element of
R, but not involving the variable
x) is
c. The pseudo remainder of
f by
g is the polynomial
h of degree less than
d in
x such that
c^(ed+1) * f = q*g + h, where
f has degree
e in
x.
i1 : R = QQ[x,y];

i2 : f = x^4
4
o2 = x
o2 : R

i3 : g = x^2*y + 13*x^2*y^4 +x*y^23*x  1
2 4 2 2
o3 = 13x y + x y + x*y  3x  1
o3 : R

i4 : (lg, cg) = topCoefficients g
2 4
o4 = (x , 13y + y)
o4 : Sequence

i5 : h = pseudoRemainder(f,g)
6 4 3 4 2 2
o5 =  27x*y + 87x*y  2x*y + 14y  27x*y + 6x*y  6y + 27x + y + 9
o5 : R

i6 : (cg^3 * f  h) % g
o6 = 0
o6 : R

i7 : q = (cg^3 * f  h) // g
2 8 2 5 6 4 2 2 3 4 2
o7 = 169x y + 26x y  13x*y + 39x*y + x y  x*y + 14y + 3x*y  6y + y

+ 9
o7 : R

i8 : cg^3*f == h + q*g
o8 = true

Caveat
There is no pseudodivision implemented, and the only way to change the notion of what the top variable is, is to change to a ring where the variables are in a different order
See also

topCoefficients  first variable and its coefficient of a polynomial or matrix
Ways to use pseudoRemainder :

pseudoRemainder(RingElement,RingElement)