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
fricas
(1) -> Z==>Integer; Q==>Fraction Z
Type: Void
fricas
CP==>DistributedMultivariatePolynomial([a,b,c], Z)
Type: Void
fricas
CF==>Fraction CP
Type: Void
fricas
P==>DistributedMultivariatePolynomial([d,n,m], CF)
Type: Void
fricas
PX==>UnivariatePolynomial('x, P)
Type: Void
fricas
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
fricas
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))))
fricas
r:PX := p(x) - d*q(y)
fricas
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.
fricas
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.
fricas
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))))
fricas
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))))
fricas
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.
fricas
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))))
fricas
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.))
fricas
solve(ecoeffs, [d,n,m]);
Type: List(List(Equation(Fraction(Polynomial(Integer)))))
===============================================================
fricas
---- Ordered variable lists.
Poly_to_Gauss:=[d,n,m]
\begin{equation*}
\label{eq9}\left[ d , \: n , \: m \right]
?\end{equation*}
fricas
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)))))
fricas
----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))
fricas
---- 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)
fricas
---- 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))
fricas
---- 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))
fricas
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)))))
fricas
---- 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))
fricas
---- 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
fricas
---- Scalar matrix
Sc(m) == diagonalMatrix [1,m,m^2]
Type: Void
fricas
---- Now define transform in matrix form
D := corg -(cgauss * Pa(n) * Sc(m))
fricas
Compiling function Pa with type Variable(n) -> Matrix(Polynomial(
Integer))fricas
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))
fricas
---- 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)))
fricas
---- 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)))))
fricas
---- 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))
fricas
Z==>Integer; Q==>Fraction Z
Type: Void
fricas
CP==>DistributedMultivariatePolynomial([a,b,c,u,v,w], Z)
Type: Void
fricas
CF==>Fraction CP
Type: Void
fricas
P==>DistributedMultivariatePolynomial([d,n,m], CF)
Type: Void
fricas
PX==>UnivariatePolynomial('x, P)
Type: Void
fricas
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
fricas
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
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,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
fricas
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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