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last edited 9 years ago by rrogers |
Edit detail for ExampleGroebner revision 8 of 8
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Editor: rrogers
Time: 2014/12/04 19:09:51 GMT+0 |
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| 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
 |
 |
 |
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,
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
 | (1) |
Type: UnivariatePolynomial(x,
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}](images/469805771725095356-16.0px.png "\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) |
Type: UnivariatePolynomial(x,
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]](images/8245104986464204879-16.0px.png "\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,
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]](images/3860665239212660254-16.0px.png "\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,
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]](images/7841115730631015885-16.0px.png "\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]](images/7052822242896589086-16.0px.png "\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]](images/4028224099335330876-16.0px.png "\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]](images/4481078259708622708-16.0px.png "\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]](images/4621265188140307077-16.0px.png "\label{eq9}\left[ d , \: n , \: m \right]") | (9) |
fricas
Gauss_to_Poly:=[x,y, a, b, c]
![\label{eq10}\left[ x , \:{{m \ x}+ n}, \: a , \: b , \: c \right]](images/3593012778196715055-16.0px.png "\label{eq10}\left[ x , \:{{m \ x}+ n}, \: a , \: b , \: c \right]") | (10) |
Type: List(UnivariatePolynomial(x,
fricas
----coefficient arrays. corg := d* matrix [[c,b, a]]
 | (11) |
Type: Matrix(Polynomial(Integer))
fricas
---- Explicit target cgauss := matrix [[0,1, -1]]
 | (12) |
Type: Matrix(Integer)
fricas
---- Generalized target ctar := matrix [[w,v, u]]
 | (13) |
Type: Matrix(Polynomial(Integer))
fricas
---- polynomial basis arrays. xorg := matrix ([[1,x, x^2]])
 | (14) |
Type: Matrix(Polynomial(Integer))
fricas
xgauss := matrix([[1,y, y^2]])
 | (15) |
Type: Matrix(UnivariatePolynomial(x,
fricas
---- Example row(corg * transpose(xorg),1)
![\label{eq16}\left[{{a \ d \ {{x}^{2}}}+{b \ d \ x}+{c \ d}}\right]](images/223290738399936045-16.0px.png "\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)) | (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]](images/7678919746467925014-16.0px.png "\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]](images/4647194367839264252-16.0px.png "\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))
 | (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,
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
| (1) |
Type: UnivariatePolynomial(x,
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) )))
| (2) |
Type: UnivariatePolynomial(x,
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](images/8532651621909741844-16.0px.png "\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) |
Type: UnivariatePolynomial(x,
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]](images/1688423701865309020-16.0px.png "\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,
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]](images/8698227003878611606-16.0px.png "\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,
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]](images/6700873344834907528-16.0px.png "\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,
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]](images/4024649792941612684-16.0px.png "\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,
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]](images/1057033562557099208-16.0px.png "\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]](images/1112376277526924279-16.0px.png "\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]](images/3475673540610227677-16.0px.png "\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]](images/1667856006545328170-16.0px.png "\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]](images/2498859117823863816-16.0px.png "\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]](images/6092855957573352995-16.0px.png "\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}](images/3824381219890199520-16.0px.png "\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]](images/3216506723505631168-16.0px.png "\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))))