MathAction SandBoxMultivariatePolynomialDivision

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A tutorial written by Angelos Mantzaflaris (webpage: http://users.uoa.gr/~amantzaf/proje.html )

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

(currently only in Greek)

contains the following routine:

fricas

-- 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:

fricas

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

$\label{eq1}\hbox{\axiomType{DistributedMultivariatePolynomial}\ } ([ x , y , z ] , \hbox{\axiomType{Fraction}\ } (\hbox{\axiomType{Integer}\ }))$

(1)

Type: Type

fricas

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

$\label{eq2}-{2 \ {{x}^{12}}}+{4 \ {{x}^{2}}\ {{y}^{4}}}$

(2)

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

fricas

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

$\label{eq3}{{x}^{2}}-{{y}^{11}}$

(3)

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

fricas

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

$\label{eq4}{3 \ {{x}^{14}}\ {{y}^{10}}}+{5 \ x \ {{y}^{3}}}-{6 \ {{y}^{7}}}+ 1$

(4)

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

fricas

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.

$\label{eq5}\begin{array}{@{}l} \displaystyle \left[{quotient ={\left[ -{{3 \over 2}\ {{x}^{2}}\ {{y}^{10}}}, \:{{6 \ {{x}^{2}}\ {{y}^{14}}}+{6 \ {{y}^{25}}}}\right]}}, \: \right. \ \ \displaystyle \left.{remainder ={{5 \ x \ {{y}^{3}}}+{6 \ {{y}^{36}}}-{6 \ {{y}^{7}}}+ 1}}\right]$

(5)

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

fricas

mpolydiv(f, [d2,d1])

$\label{eq6}\begin{array}{@{}l} \displaystyle \left[{ \begin{array}{@{}l} \displaystyle quotient ={ \begin{array}{@{}l} \displaystyle \left[{{3 \ {{x}^{12}}\ {{y}^{10}}}+{3 \ {{x}^{10}}\ {{y}^{21}}}+{3 \ {{x}^{8}}\ {{y}^{32}}}+{3 \ {{x}^{6}}\ {{y}^{43}}}+{3 \ {{x}^{4}}\ {{y}^{54}}}+{3 \ {{x}^{2}}\ {{y}^{65}}}+{3 \ {{y}^{76}}}}, \right. \ \ \displaystyle \left.\: 0 \right]$

(6)

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

fricas

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

$\label{eq7}{3 \ {{x}^{14}}\ {{y}^{10}}}+{5 \ x \ {{y}^{3}}}-{6 \ {{y}^{7}}}+ 1$

(7)

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

Subject: Be Bold !!

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