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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
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

\label{eq1}{m \  x}+ n(1)
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))))

\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}
(2)

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

\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] 
(3)
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

\label{eq4}\left[{d -{a \over{{4 \  a \  c}-{{b}^{2}}}}}, \:{n +{{1 \over 2}\  m}+{b \over{2 \  a}}}, \:{{{m}^{2}}+{{{4 \  a \  c}-{{b}^{2}}}\over{{a}^{2}}}}\right](4)
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]

\label{eq5}\begin{array}{@{}l}
\displaystyle
\left[{{{{{\left({4 \  a \  c}-{{b}^{2}}\right)}\  d}- a}\over{{4 \  a \  c}-{{b}^{2}}}}= 0}, \:{{{{2 \  a \  n}+{a \  m}+ b}\over{2 \  a}}= 0}, \: \right.
\
\
\displaystyle
\left.{{{{{{a}^{2}}\ {{m}^{2}}}+{4 \  a \  c}-{{b}^{2}}}\over{{a}^{2}}}= 0}\right] 
(5)
Type: List(Equation(Fraction(Polynomial(Integer))))
fricas
solve(egb, [d,m,n])

\label{eq6}\left[{\left[{d ={a \over{{4 \  a \  c}-{{b}^{2}}}}}, \:{m ={{-{2 \  a \  n}- b}\over a}}, \:{{{a \ {{n}^{2}}}+{b \  n}+ c}= 0}\right]}\right](6)
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]

\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] (7)
Type: List(Equation(Fraction(Polynomial(Integer))))
fricas
solve(ecoeffs, [d,m,n])

\label{eq8}\left[{\left[{d ={a \over{{4 \  a \  c}-{{b}^{2}}}}}, \:{m ={{-{2 \  a \  n}- b}\over a}}, \:{{{a \ {{n}^{2}}}+{b \  n}+ c}= 0}\right]}\right](8)
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)))))

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

Example code --rrogers, Tue, 02 Dec 2014 23:49:41 +0000 reply
fricas
---- Ordered  variable lists.
Poly_to_Gauss:=[d,n,m]

\label{eq9}\left[ d , \: n , \: m \right](9)
Type: List(OrderedVariableList([d,n,m]))
fricas
Gauss_to_Poly:=[x,y,a,b,c]

\label{eq10}\left[ x , \:{{m \  x}+ n}, \: a , \: b , \: c \right](10)
fricas
----coefficient arrays.
corg :=  d* matrix [[c,b,a]]

\label{eq11}\left[ 
\begin{array}{ccc}
{c \  d}&{b \  d}&{a \  d}
(11)
Type: Matrix(Polynomial(Integer))
fricas
---- Explicit target
cgauss := matrix [[0, 1, -1]]

\label{eq12}\left[ 
\begin{array}{ccc}
0 & 1 & - 1 
(12)
Type: Matrix(Integer)
fricas
---- Generalized target
ctar := matrix [[w,v,u]]

\label{eq13}\left[ 
\begin{array}{ccc}
w & v & u 
(13)
Type: Matrix(Polynomial(Integer))
fricas
---- polynomial basis arrays.
xorg := matrix ([[1, x, x^2]])

\label{eq14}\left[ 
\begin{array}{ccc}
1 & x &{{x}^{2}}
(14)
Type: Matrix(Polynomial(Integer))
fricas
xgauss := matrix([[1,y,y^2]])

\label{eq15}\left[ 
\begin{array}{ccc}
1 &{{m \  x}+ n}&{{{{m}^{2}}\ {{x}^{2}}}+{2 \  n \  m \  x}+{{n}^{2}}}
(15)
fricas
---- Example
row(corg * transpose(xorg),1)

\label{eq16}\left[{{a \  d \ {{x}^{2}}}+{b \  d \  x}+{c \  d}}\right](16)
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))

\label{eq17}\left[ 
\begin{array}{ccc}
{{{n}^{2}}- n +{c \  d}}&{{2 \  m \  n}- m +{b \  d}}&{{{m}^{2}}+{a \  d}}
(17)
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, 2m n - m + b d, 2b d n + (- 4c d + 1)m - b d, 2a n - b m - a, 2 2 m + a d, (4a c - b )d - a] factorGroebnerBasis: no reducible polynomials in this basis

\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] (18)
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)

\label{eq19}\left[{\left[{d ={a \over{{4 \  a \  c}-{{b}^{2}}}}}, \:{n ={{{b \  m}+ a}\over{2 \  a}}}, \:{{{{\left({4 \  a \  c}-{{b}^{2}}\right)}\ {{m}^{2}}}+{{a}^{2}}}= 0}\right]}\right](19)
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))

\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}}
(20)
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.
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.
Type: Void
fricas
y:PX := m*x+n

\label{eq1}{m \  x}+ n(1)
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,u,v,w],Integer)))) -> 
      UnivariatePolynomial(x,DistributedMultivariatePolynomial([d,n,m],
      Fraction(DistributedMultivariatePolynomial([a,b,c,u,v,w],Integer)
      )))
fricas
Compiling function q with type 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)
      )))

\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}
(2)
fricas
s:PX := fp(x) - d*q(y)
fricas
Compiling function fp with type 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)
      )))

\label{eq3}\begin{array}{@{}l}
\displaystyle
{{\left(-{a \  d \ {{m}^{2}}}+ u \right)}\ {{x}^{2}}}+{{\left(-{2 \  a \  d \  n \  m}-{b \  d \  m}+ v \right)}\  x}-{a \  d \ {{n}^{2}}}- 
\
\
\displaystyle
{b \  d \  n}-{c \  d}+ w 
(3)
fricas
coeffs := coefficients r

\label{eq4}\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] 
(4)
Type: List(DistributedMultivariatePolynomial([d,n,m],Fraction(DistributedMultivariatePolynomial([a,b,c,u,v,w],Integer))))
fricas
fcoeffs := coefficients s

\label{eq5}\begin{array}{@{}l}
\displaystyle
\left[{-{a \  d \ {{m}^{2}}}+ u}, \:{-{2 \  a \  d \  n \  m}-{b \  d \  m}+ v}, \: \right.
\
\
\displaystyle
\left.{-{a \  d \ {{n}^{2}}}-{b \  d \  n}-{c \  d}+ w}\right] (5)
Type: List(DistributedMultivariatePolynomial([d,n,m],Fraction(DistributedMultivariatePolynomial([a,b,c,u,v,w],Integer))))
fricas
gb := groebner coeffs

\label{eq6}\left[{d -{a \over{{4 \  a \  c}-{{b}^{2}}}}}, \:{n +{{1 \over 2}\  m}+{b \over{2 \  a}}}, \:{{{m}^{2}}+{{{4 \  a \  c}-{{b}^{2}}}\over{{a}^{2}}}}\right](6)
Type: List(DistributedMultivariatePolynomial([d,n,m],Fraction(DistributedMultivariatePolynomial([a,b,c,u,v,w],Integer))))
fricas
fgb := groebner fcoeffs

\label{eq7}\begin{array}{@{}l}
\displaystyle
\left[{d +{{-{4 \  a \  u \  w}+{a \ {{v}^{2}}}}\over{{4 \  a \  c \  u}-{{{b}^{2}}\  u}}}}, \:{n -{{v \over{2 \  u}}\  m}+{b \over{2 \  a}}}, \: \right.
\
\
\displaystyle
\left.{{{m}^{2}}+{{-{4 \  a \  c \ {{u}^{2}}}+{{{b}^{2}}\ {{u}^{2}}}}\over{{4 \ {{a}^{2}}\  u \  w}-{{{a}^{2}}\ {{v}^{2}}}}}}\right] (7)
Type: List(DistributedMultivariatePolynomial([d,n,m],Fraction(DistributedMultivariatePolynomial([a,b,c,u,v,w],Integer))))
fricas
egb: List Equation Fraction Polynomial Z := [p=0 for p in gb]

\label{eq8}\begin{array}{@{}l}
\displaystyle
\left[{{{{{\left({4 \  a \  c}-{{b}^{2}}\right)}\  d}- a}\over{{4 \  a \  c}-{{b}^{2}}}}= 0}, \:{{{{2 \  a \  n}+{a \  m}+ b}\over{2 \  a}}= 0}, \: \right.
\
\
\displaystyle
\left.{{{{{{a}^{2}}\ {{m}^{2}}}+{4 \  a \  c}-{{b}^{2}}}\over{{a}^{2}}}= 0}\right] 
(8)
Type: List(Equation(Fraction(Polynomial(Integer))))
fricas
fegb:  List Equation Fraction Polynomial Z := [p=0 for p in fgb]

\label{eq9}\begin{array}{@{}l}
\displaystyle
\left[{{{-{4 \  a \  u \  w}+{a \ {{v}^{2}}}+{{\left({4 \  a \  c}-{{b}^{2}}\right)}\  d \  u}}\over{{\left({4 \  a \  c}-{{b}^{2}}\right)}\  u}}= 0}, \: \right.
\
\
\displaystyle
\left.{{{-{a \  m \  v}+{{\left({2 \  a \  n}+ b \right)}\  u}}\over{2 \  a \  u}}= 0}, \: \right.
\
\
\displaystyle
\left.{{{{4 \ {{a}^{2}}\ {{m}^{2}}\  u \  w}-{{{a}^{2}}\ {{m}^{2}}\ {{v}^{2}}}+{{\left(-{4 \  a \  c}+{{b}^{2}}\right)}\ {{u}^{2}}}}\over{{4 \ {{a}^{2}}\  u \  w}-{{{a}^{2}}\ {{v}^{2}}}}}= 0}\right] 
(9)
Type: List(Equation(Fraction(Polynomial(Integer))))
fricas
ecoeffs: List Equation Fraction Polynomial Z := [p=0 for p in coeffs]

\label{eq10}\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] (10)
Type: List(Equation(Fraction(Polynomial(Integer))))
fricas
fecoeffs: List Equation Fraction Polynomial Z := [p=0 for p in fcoeffs]

\label{eq11}\begin{array}{@{}l}
\displaystyle
\left[{{u -{a \  d \ {{m}^{2}}}}= 0}, \:{{v -{2 \  a \  d \  m \  n}-{b \  d \  m}}= 0}, \: \right.
\
\
\displaystyle
\left.{{w -{a \  d \ {{n}^{2}}}-{b \  d \  n}-{c \  d}}= 0}\right] (11)
Type: List(Equation(Fraction(Polynomial(Integer))))
fricas
cc:=solve(egb, [d,n,m]);
Type: List(List(Equation(Fraction(Polynomial(Integer)))))
fricas
cc.1

\label{eq12}\left[{d ={a \over{{4 \  a \  c}-{{b}^{2}}}}}, \:{n ={{-{a \  m}- b}\over{2 \  a}}}, \:{{{{{a}^{2}}\ {{m}^{2}}}+{4 \  a \  c}-{{b}^{2}}}= 0}\right](12)
Type: List(Equation(Fraction(Polynomial(Integer))))
fricas
dd:=solve(ecoeffs, [d,n,m]);
Type: List(List(Equation(Fraction(Polynomial(Integer)))))
fricas
dd.1

\label{eq13}\left[{d ={a \over{{4 \  a \  c}-{{b}^{2}}}}}, \:{n ={{-{a \  m}- b}\over{2 \  a}}}, \:{{{{{a}^{2}}\ {{m}^{2}}}+{4 \  a \  c}-{{b}^{2}}}= 0}\right](13)
Type: List(Equation(Fraction(Polynomial(Integer))))
fricas
fcc:=solve(fegb,[d,n,m]);
Type: List(List(Equation(Fraction(Polynomial(Integer)))))
fricas
fcc.1

\label{eq14}\begin{array}{@{}l}
\displaystyle
\left[{d ={{{4 \  a \  u \  w}-{a \ {{v}^{2}}}}\over{{\left({4 \  a \  c}-{{b}^{2}}\right)}\  u}}}, \: \right.
\
\
\displaystyle
\left.{
\begin{array}{@{}l}
\displaystyle
n ={{\left(
\begin{array}{@{}l}
\displaystyle
{{\left({{\left(-{4 \ {{a}^{2}}\ {{m}^{2}}}+{4 \ {{a}^{2}}\  m}\right)}\  u \  v}-{4 \  a \  b \ {{u}^{2}}}\right)}\  w}+ \
\
\displaystyle
{{\left({{{a}^{2}}\ {{m}^{2}}}-{{{a}^{2}}\  m}\right)}\ {{v}^{3}}}+{a \  b \  u \ {{v}^{2}}}+ 
\
\
\displaystyle
{{\left({4 \  a \  c}-{{b}^{2}}\right)}\ {{u}^{2}}\  v}
(14)
Type: List(Equation(Fraction(Polynomial(Integer))))
fricas
fdd:=solve(fecoeffs, [d,n,m]);
Type: List(List(Equation(Fraction(Polynomial(Integer)))))
fricas
fdd.1

\label{eq15}\begin{array}{@{}l}
\displaystyle
\left[{d ={{{4 \  a \  u \  w}-{a \ {{v}^{2}}}}\over{{\left({4 \  a \  c}-{{b}^{2}}\right)}\  u}}}, \:{n ={{{a \  m \  v}-{b \  u}}\over{2 \  a \  u}}}, \: \right.
\
\
\displaystyle
\left.{{{4 \ {{a}^{2}}\ {{m}^{2}}\  u \  w}-{{{a}^{2}}\ {{m}^{2}}\ {{v}^{2}}}+{{\left(-{4 \  a \  c}+{{b}^{2}}\right)}\ {{u}^{2}}}}= 0}\right] 
(15)
Type: List(Equation(Fraction(Polynomial(Integer))))




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