pseudoRemainder -- compute the pseudo-remainder

Synopsis

Usage:

pseudoRemainder(f,g)
Inputs:
- f, a ring element
- g, a ring element, in the same polynomial ring R as f
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^(e-d+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^2-3*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 pseudo-division 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

Ways to use pseudoRemainder :

pseudoRemainder(RingElement,RingElement)

For the programmer