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ExampleGroebner

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Editor: rrogers Time: 2014/12/04 19:09:51 GMT+0
Note: edit test

added:

From rrogers Thu Dec 4 19:09:49 +0000 2014
From: rrogers
Date: Thu, 04 Dec 2014 19:09:49 +0000
Subject: edit test
Message-ID: <20141204190949+0000@axiom-wiki.newsynthesis.org>

\begin{axiom}
Z==>Integer; Q==>Fraction Z
CP==>DistributedMultivariatePolynomial([a,b,c,u,v,w], Z)
CF==>Fraction CP
P==>DistributedMultivariatePolynomial([d,n,m], CF)
PX==>UnivariatePolynomial('x, P)
p(x:PX):PX == x*(1-x)
fp(x:PX):PX == u*x^2+v*x+w
q(y:PX):PX == a*y^2+b*y+c;
y:PX := m*x+n

r:PX := p(x) - d*q(y)
s:PX := fp(x) - d*q(y)

coeffs := coefficients r
fcoeffs := coefficients s
gb := groebner coeffs
fgb := groebner fcoeffs
egb: List Equation Fraction Polynomial Z := [p=0 for p in gb]
fegb:  List Equation Fraction Polynomial Z := [p=0 for p in fgb]
ecoeffs: List Equation Fraction Polynomial Z := [p=0 for p in coeffs]
fecoeffs: List Equation Fraction Polynomial Z := [p=0 for p in fcoeffs]


cc:=solve(egb, [d,n,m]);
cc.1
dd:=solve(ecoeffs, [d,n,m]);
dd.1
fcc:=solve(fegb,[d,n,m]);
fcc.1

fdd:=solve(fecoeffs, [d,n,m]);
fdd.1


\end{axiom}

Application of Groebner Bases

Problem

Let $$p(x) = - x^2 + x, \qquad q(y) = a y^2 + b y + c.$$ Find d, m, n (depending on the coefficients a,b,c of q) such that for the transformaton $$y = m x + n$$ it holds $$p(x) = d q(m x + n).$$

Setup of the problem

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(1) -> Z==>Integer; Q==>Fraction Z

Type: Void

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CP==>DistributedMultivariatePolynomial([a,b,c], Z)

Type: Void

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CF==>Fraction CP

Type: Void

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P==>DistributedMultivariatePolynomial([d,n,m], CF)

Type: Void

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PX==>UnivariatePolynomial('x, P)

Type: Void

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p(x:PX):PX == x*(1-x)

   Function declaration p : UnivariatePolynomial(x,
      DistributedMultivariatePolynomial([d,n,m],Fraction(
      DistributedMultivariatePolynomial([a,b,c],Integer)))) -> 
      UnivariatePolynomial(x,DistributedMultivariatePolynomial([d,n,m],
      Fraction(DistributedMultivariatePolynomial([a,b,c],Integer)))) 
      has been added to workspace.

Type: Void

fricas

q(y:PX):PX == a*y^2+b*y+c

   Function declaration q : UnivariatePolynomial(x,
      DistributedMultivariatePolynomial([d,n,m],Fraction(
      DistributedMultivariatePolynomial([a,b,c],Integer)))) -> 
      UnivariatePolynomial(x,DistributedMultivariatePolynomial([d,n,m],
      Fraction(DistributedMultivariatePolynomial([a,b,c],Integer)))) 
      has been added to workspace.

Type: Void

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y:PX := m*x+n

\begin{equation} \label{eq1}{m \ x}+ n\end{equation}

Type: UnivariatePolynomial(x,DistributedMultivariatePolynomial?([d,n,m],Fraction(DistributedMultivariatePolynomial?([a,b,c],Integer))))

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r:PX := p(x) - d*q(y)

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Compiling function p with type UnivariatePolynomial(x,
      DistributedMultivariatePolynomial([d,n,m],Fraction(
      DistributedMultivariatePolynomial([a,b,c],Integer)))) -> 
      UnivariatePolynomial(x,DistributedMultivariatePolynomial([d,n,m],
      Fraction(DistributedMultivariatePolynomial([a,b,c],Integer))))

fricas

Compiling function q with type UnivariatePolynomial(x,
      DistributedMultivariatePolynomial([d,n,m],Fraction(
      DistributedMultivariatePolynomial([a,b,c],Integer)))) -> 
      UnivariatePolynomial(x,DistributedMultivariatePolynomial([d,n,m],
      Fraction(DistributedMultivariatePolynomial([a,b,c],Integer))))

\begin{equation} \label{eq2}\begin{array}{@{}l} \displaystyle {{\left(-{a \ d \ {{m}^{2}}}- 1 \right)}\ {{x}^{2}}}+{{\left(-{2 \ a \ d \ n \ m}-{b \ d \ m}+ 1 \right)}\ x}-{a \ d \ {{n}^{2}}}- \ \ \displaystyle {b \ d \ n}-{c \ d} \end{array} \end{equation}

Type: UnivariatePolynomial(x,DistributedMultivariatePolynomial?([d,n,m],Fraction(DistributedMultivariatePolynomial?([a,b,c],Integer))))

Compute the solution

We must first extract the coefficients, since each coefficient of any power of x must vanish if the polynomial r is identically 0.

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coeffs := coefficients r

\begin{equation*} \label{eq3}\begin{array}{@{}l} \displaystyle \left[{-{a \ d \ {{m}^{2}}}- 1}, \:{-{2 \ a \ d \ n \ m}-{b \ d \ m}+ 1}, \: \right. \ \ \displaystyle \left.{-{a \ d \ {{n}^{2}}}-{b \ d \ n}-{c \ d}}\right] \end{array} \end{equation*}

Type: List(DistributedMultivariatePolynomial?([d,n,m],Fraction(DistributedMultivariatePolynomial?([a,b,c],Integer))))

Now we compute a Groebner basis and then solve for the respective variables.

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gb := groebner coeffs

\begin{equation*} \label{eq4}\begin{array}{@{}l} \displaystyle \left[{d -{\frac{a}{{4 \ a \ c}-{{b}^{2}}}}}, \:{n +{{\frac{1}{2}}\ m}+{\frac{b}{2 \ a}}}, \: \right. \ \ \displaystyle \left.{{{m}^{2}}+{\frac{{4 \ a \ c}-{{b}^{2}}}{{a}^{2}}}}\right] \end{array} \end{equation*}

Type: List(DistributedMultivariatePolynomial?([d,n,m],Fraction(DistributedMultivariatePolynomial?([a,b,c],Integer))))

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egb: List Equation Fraction Polynomial Z := [p=0 for p in gb]

\begin{equation*} \label{eq5}\begin{array}{@{}l} \displaystyle \left[{{\frac{{{\left({4 \ a \ c}-{{b}^{2}}\right)}\ d}- a}{{4 \ a \ c}-{{b}^{2}}}}= 0}, \: \right. \ \ \displaystyle \left.{{\frac{{2 \ a \ n}+{a \ m}+ b}{2 \ a}}= 0}, \:{{\frac{{{{a}^{2}}\ {{m}^{2}}}+{4 \ a \ c}-{{b}^{2}}}{{a}^{2}}}= 0}\right] \end{array} \end{equation*}

Type: List(Equation(Fraction(Polynomial(Integer))))

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solve(egb, [d,m,n])

\begin{equation*} \label{eq6}\begin{array}{@{}l} \displaystyle \left[ \left[{d ={\frac{a}{{4 \ a \ c}-{{b}^{2}}}}}, \:{m ={\frac{-{2 \ a \ n}- b}{a}}}, \: \right. \ \ \displaystyle \left.{{{a \ {{n}^{2}}}+{b \ n}+ c}= 0}\right] \right] \end{array} \end{equation*}

Type: List(List(Equation(Fraction(Polynomial(Integer)))))

In fact, solve is powerful enough so that it is unnecessary to call the Buchberger algorithm explicitly.

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ecoeffs: List Equation Fraction Polynomial Z := [p=0 for p in coeffs]

\begin{equation*} \label{eq7}\begin{array}{@{}l} \displaystyle \left[{{-{a \ d \ {{m}^{2}}}- 1}= 0}, \:{{-{2 \ a \ d \ m \ n}-{b \ d \ m}+ 1}= 0}, \: \right. \ \ \displaystyle \left.{{-{a \ d \ {{n}^{2}}}-{b \ d \ n}-{c \ d}}= 0}\right] \end{array} \end{equation*}

Type: List(Equation(Fraction(Polynomial(Integer))))

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solve(ecoeffs, [d,m,n])

\begin{equation*} \label{eq8}\begin{array}{@{}l} \displaystyle \left[ \left[{d ={\frac{a}{{4 \ a \ c}-{{b}^{2}}}}}, \:{m ={\frac{-{2 \ a \ n}- b}{a}}}, \: \right. \ \ \displaystyle \left.{{{a \ {{n}^{2}}}+{b \ n}+ c}= 0}\right] \right] \end{array} \end{equation*}

Type: List(List(Equation(Fraction(Polynomial(Integer)))))

Of course, the result depends on the order of the variables given to the solve command.

((Unfortunately, the axiom-wiki does not properly show the result, so we have added a semicolon to suppress the output.))

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solve(ecoeffs, [d,n,m]);

Type: List(List(Equation(Fraction(Polynomial(Integer)))))

===============================================================

Example code --rrogers, Tue, 02 Dec 2014 23:49:41 +0000 reply

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---- Ordered  variable lists.
Poly_to_Gauss:=[d,n,m]

\begin{equation*} \label{eq9}\left[ d , \: n , \: m \right]?\end{equation*}

Type: List(OrderedVariableList([d,n,m]))

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Gauss_to_Poly:=[x,y,a,b,c]

\begin{equation*} \label{eq10}\left[ x , \:{{m \ x}+ n}, \: a , \: b , \: c \right]?\end{equation*}

Type: List(UnivariatePolynomial(x,DistributedMultivariatePolynomial?([d,n,m],Fraction(DistributedMultivariatePolynomial?([a,b,c],Integer)))))

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----coefficient arrays.
corg :=  d* matrix [[c,b,a]]

\begin{equation*} \label{eq11}\left[ \begin{array}{ccc} {c \ d}&{b \ d}&{a \ d} \end{array} \right]\end{equation*}

Type: Matrix(Polynomial(Integer))

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---- Explicit target
cgauss := matrix [[0, 1, -1]]

\begin{equation*} \label{eq12}\left[ \begin{array}{ccc} 0 & 1 & - 1 \end{array} \right]\end{equation*}

Type: Matrix(Integer)

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---- Generalized target
ctar := matrix [[w,v,u]]

\begin{equation*} \label{eq13}\left[ \begin{array}{ccc} w & v & u \end{array} \right]\end{equation*}

Type: Matrix(Polynomial(Integer))

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---- polynomial basis arrays.
xorg := matrix ([[1, x, x^2]])

\begin{equation*} \label{eq14}\left[ \begin{array}{ccc} 1 & x &{{x}^{2}} \end{array} \right]\end{equation*}

Type: Matrix(Polynomial(Integer))

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xgauss := matrix([[1,y,y^2]])

\begin{equation*} \label{eq15}\left[ \begin{array}{ccc} 1 &{{m \ x}+ n}&{{{{m}^{2}}\ {{x}^{2}}}+{2 \ n \ m \ x}+{{n}^{2}}} \end{array} \right]\end{equation*}

Type: Matrix(UnivariatePolynomial(x,DistributedMultivariatePolynomial?([d,n,m],Fraction(DistributedMultivariatePolynomial?([a,b,c],Integer)))))

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---- Example
row(corg * transpose(xorg),1)

\begin{equation*} \label{eq16}\left[{{a \ d \ {{x}^{2}}}+{b \ d \ x}+{c \ d}}\right]?\end{equation*}

Type: Vector(Polynomial(Integer))

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----  Translation matrix Pascal Pa(n) for 3x3 case
----  see Aceto below for references.
Pa(n) == matrix [[1,0,0],[n,1,0],[n^2, 2*n,1]]

Type: Void

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---- Scalar matrix
Sc(m) == diagonalMatrix [1,m,m^2]

Type: Void

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---- Now define transform in matrix form
D := corg -(cgauss * Pa(n) * Sc(m))

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Compiling function Pa with type Variable(n) -> Matrix(Polynomial(
      Integer))

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Compiling function Sc with type Variable(m) -> Matrix(Polynomial(
      Integer))

\begin{equation*} \label{eq17}\left[ \begin{array}{ccc} {{{n}^{2}}- n +{c \ d}}&{{2 \ m \ n}- m +{b \ d}}&{{{m}^{2}}+{a \ d}} \end{array} \right]\end{equation*}

Type: Matrix(Polynomial(Integer))

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---- Now we do a more realistic solve in two steps
---- Step one disallow silly answers
E:=groebnerFactorize(row(D,1),[b*d,m,a,b^2-3*a*c],true)

   we found a groebner basis and check whether it
   contains reducible polynomials
   [1]
   factorGroebnerBasis: no reducible polynomials in this basis
   we found a groebner basis and check whether it
   contains reducible polynomials
     2
   [n  - n + c d, 2 m n - m + b d, 2 b d n + (- 4 c d + 1)m - b d,
                      2                  2
    2 a n - b m - a, m  + a d, (4 a c - b )d - a]
   factorGroebnerBasis: no reducible polynomials in this basis

\begin{equation*} \label{eq18}\begin{array}{@{}l} \displaystyle \left[{ \begin{array}{@{}l} \displaystyle \left[{{{n}^{2}}- n +{c \ d}}, \:{{2 \ m \ n}- m +{b \ d}}, \: \right. \ \ \displaystyle \left.{{2 \ b \ d \ n}+{{\left(-{4 \ c \ d}+ 1 \right)}\ m}-{b \ d}}, \:{{2 \ a \ n}-{b \ m}- a}, \:{{{m}^{2}}+{a \ d}}, \: \right. \ \ \displaystyle \left.{{{\left({4 \ a \ c}-{{b}^{2}}\right)}\ d}- a}\right] \end{array} }, \right. \ \ \displaystyle \left.\:{\left[ 1 \right]?}\right] \end{array} \end{equation*}

Type: List(List(Polynomial(Integer)))

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----  and clean it up (a lot).  I wish these two steps could be one!
solve(E.1,Poly_to_Gauss)

\begin{equation*} \label{eq19}\begin{array}{@{}l} \displaystyle \left[ \left[{d ={\frac{a}{{4 \ a \ c}-{{b}^{2}}}}}, \:{n ={\frac{{b \ m}+ a}{2 \ a}}}, \: \right. \ \ \displaystyle \left.{{{{\left({4 \ a \ c}-{{b}^{2}}\right)}\ {{m}^{2}}}+{{a}^{2}}}= 0}\right] \right] \end{array} \end{equation*}

Type: List(List(Equation(Fraction(Polynomial(Integer)))))

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---- Now lets test the reasonableness the width to start with is
---- 2*sqrt(b^2-4*a*c)/(2*a)  which the left hand term yields.  There is a sign ambiguity
---- corresponding to whether the source quadratic is to the left or right.
---- I could swap n,m in solve() but then the n term (left hand one) is more obscure
---- Knowing the width m we can compute moving the center to 1/2 (for x*(1-x))
---- It should amount to -b/(2*a)+1/2 
---- and in fact that is the answer n= m(scale factor)*(b/2a)+1/2
---- d is required and in English is a "normalizing factor"

----General formulation
Dorg := corg -(ctar * Pa(n) * Sc(m))

\begin{equation*} \label{eq20}\left[ \begin{array}{ccc} {- w -{n \ v}-{{{n}^{2}}\ u}+{c \ d}}&{-{m \ v}-{2 \ m \ n \ u}+{b \ d}}&{-{{{m}^{2}}\ u}+{a \ d}} \end{array} \right]\end{equation*}

Type: Matrix(Polynomial(Integer))

edit test --rrogers, Thu, 04 Dec 2014 19:09:49 +0000 reply

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Z==>Integer; Q==>Fraction Z

Type: Void

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CP==>DistributedMultivariatePolynomial([a,b,c,u,v,w], Z)

Type: Void

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CF==>Fraction CP

Type: Void

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P==>DistributedMultivariatePolynomial([d,n,m], CF)

Type: Void

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PX==>UnivariatePolynomial('x, P)

Type: Void

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p(x:PX):PX == x*(1-x)

   Function declaration p : UnivariatePolynomial(x,
      DistributedMultivariatePolynomial([d,n,m],Fraction(
      DistributedMultivariatePolynomial([a,b,c,u,v,w],Integer)))) -> 
      UnivariatePolynomial(x,DistributedMultivariatePolynomial([d,n,m],
      Fraction(DistributedMultivariatePolynomial([a,b,c,u,v,w],Integer)
      ))) has been added to workspace.
   Compiled code for p has been cleared.
   1 old definition(s) deleted for function or rule p

Type: Void

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fp(x:PX):PX == u*x^2+v*x+w

   Function declaration fp : UnivariatePolynomial(x,
      DistributedMultivariatePolynomial([d,n,m],Fraction(
      DistributedMultivariatePolynomial([a,b,c,u,v,w],Integer)))) -> 
      UnivariatePolynomial(x,DistributedMultivariatePolynomial([d,n,m],
      Fraction(DistributedMultivariatePolynomial([a,b,c,u,v,w],Integer)
      ))) has been added to workspace.

Type: Void

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q(y:PX):PX == a*y^2+b*y+c;

   Function declaration q : UnivariatePolynomial(x,
      DistributedMultivariatePolynomial([d,n,m],Fraction(
      DistributedMultivariatePolynomial([a,b,c,u,v,w],Integer)))) -> 
      UnivariatePolynomial(x,DistributedMultivariatePolynomial([d,n,m],
      Fraction(DistributedMultivariatePolynomial([a,b,c,u,v,w],Integer)
      ))) has been added to workspace.
   Compiled code for q has been cleared.
   1 old definition(s) deleted for function or rule q

Type: Void

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y:PX := m*x+n

   You cannot declare y to be of type UnivariatePolynomial(x,
      DistributedMultivariatePolynomial([d,n,m],Fraction(
      DistributedMultivariatePolynomial([a,b,c,u,v,w],Integer)))) 
      because either the declared type of y or the type of the value of
      y is different from UnivariatePolynomial(x,
      DistributedMultivariatePolynomial([d,n,m],Fraction(
      DistributedMultivariatePolynomial([a,b,c,u,v,w],Integer)))) .

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