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SandBoxMultivariatePolynomialDivision

Edit detail for SandBoxMultivariatePolynomialDivision revision 2 of 3

	1 2 3
Editor: Bill Page Time: 2008/07/22 06:23:26 GMT-7
Note: new

changed:
-      if normalForm ( leadingMonomial (f),[ lms .j ])=0 then
-        t := ( leadingMonomial f/lms.j):: lex
-        Q.j := Q.j + t
-        f := f - t*D.j
-        flag := true
-      else
-        j:= j+1
    if normalForm ( leadingMonomial (f),[ lms .j ])=0 then
      t := ( leadingMonomial f/lms.j):: lex
      Q.j := Q.j + t
      f := f - t*D.j
      flag := true
    else
      j:= j+1

changed:
-.quotient 1 * d2 + %.quotient 2 * d1 + %.remainder
%.quotient 1 * d2 + %.quotient 2 * d1 + %.remainder

A tutorial written by Angelos Mantzaflaris (currently on in Greek)

http://eclass.uoa.gr/courses/MATH117/document/Simeioseis_axiom.pdf

contains the following routine:

axiom-- Multivariate Polynomial Division
mpolydiv (f,D)==
  lex := typeOf (f)
  Q:=[ 0:: lex for j in D ]
  r:= 0:: lex
  m:= #D
  lms :=[ leadingMonomial (D.j) for j in 1..m]
  j:= 1
  flag := false
  while f ~= 0 repeat
    if j>m then
      j:= 1
      if flag = false then
        t:= leadingMonomial f
        r:= r + t
        f:= f - t
      flag := false
    if normalForm ( leadingMonomial (f),[ lms .j ])=0 then
      t := ( leadingMonomial f/lms.j):: lex
      Q.j := Q.j + t
      f := f - t*D.j
      flag := true
    else
      j:= j+1
  rec := Record ( quotient : LIST lex , remainder : lex)
  return [Q,r]:: rec

Type: Void

Here is an example of it's use:

axiom
lex:=DMP([x,y,z], FRAC INT)

(1)

Type: Domain

axiom
d1: lex := -2* x^12 + 4* x^2 * y^4

(2)

Type: DistributedMultivariatePolynomial?([x,y,z]?,Fraction Integer)

axiom
d2: lex := x^2 - y^11

(3)

Type: DistributedMultivariatePolynomial?([x,y,z]?,Fraction Integer)

axiom
f :lex := 3*x^14*y^10 + 5*x*y^3 -6*y^7 + 1

(4)

Type: DistributedMultivariatePolynomial?([x,y,z]?,Fraction Integer)

axiom
mpolydiv(f, [d1,d2])

   Cannot compile conversion for types involving local variables. In
      particular, could not compile the expression involving :: lex
   FriCAS will attempt to step through and interpret the code.

(5)

Type: Record(quotient: List DistributedMultivariatePolynomial?([x,y,z]?,Fraction Integer),remainder: DistributedMultivariatePolynomial?([x,y,z]?,Fraction Integer))

axiom
mpolydiv(f, [d2,d1])

(6)

Type: Record(quotient: List DistributedMultivariatePolynomial?([x,y,z]?,Fraction Integer),remainder: DistributedMultivariatePolynomial?([x,y,z]?,Fraction Integer))

axiom
%.quotient 1 * d2 + %.quotient 2 * d1 + %.remainder

(7)

Type: DistributedMultivariatePolynomial?([x,y,z]?,Fraction Integer)