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how to solve without free variable t...

Maple
`g1 := x^2-w*y;g2 := x*y-w*z;g3 := y^2-x*z;S12 := SPolynomial(g1, g2, plex(x, y, z, w));eq1:= S12 = u1*g1 + u2*g2 + u3*g3;with(LinearAlgebra):T:=GenerateMatrix([eq1],[u1,u2, u3],augmented);S:=LinearSolve(T,free=t);(u1,u2,u3) should be (0,0,-w);`
` `
`the goal is find a u1,u2,u3 to make eq1 equal S12, where u1,u2,u3 should be in terms of x y z w`

Which command can output lead power of p...

Maple
`g3 := y^2-x*z;X3 := LeadingPowerOfProduct(g3, plex(x,y,z,w));X3 should be y^2`

how to find solution minimize the cost f...

Maple

3*rho1 - 2*rho2 + rho3 - rho4 = -1

4*rho1 +   rho2 - rho3        = 5

original without cost function:

with(Groebner):
K := {y1-(x1^3)*(x2^4),y2-(x2^(1+2))*(w^2),y3-(x1^(1+1))*(w^1),y4-(x2^1)*w,(y1^1000)*(y2^1)*(y3^1)*(y4^100)- x1*x2*w + 1};
G := Basis(K, plex(x1, x2, w, y1, y2, y3, y4));
Reduce((x2^(5+1))*(w^1), G, plex(x1, x2, w, y1, y2, y3, y4));

after have cost function 1000*rho1 + rho2...

how to find use all possible reduce resu...

with(Groebner):
K := {y1-(x1^3)*(x2^4),y2-(x2^(1+2))*(w^2),y3-(x1^(1+1))*(w^1),y4-(x2^1)*w,x1*x2*w - 1};
G := Basis(K, plex(x1, x2, w, y1, y2, y3, y4));
Reduce((x2^(5+1))*(w^1), G, plex(x1, x2, w, y1, y2, y3, y4));
(1,1,0,2)

subs({rho1=1,rho2=1,rho3=0,rho4=2},3*rho1 - 2*rho2 + rho3 - rho4);
subs({rho1=1,rho2=1,rho3=0,rho4=2},4*rho1 +   rho2 - rho3       );

can also reduce with G8,G7

R1 := Reduce((x2^(5+1))*(w^1), {G[7...

How to summation and simplify q calculus...

Maple

i am doing a Q Laplace and finally summation it

however, i find an AccurateQSummation which is not like Summation

Should i use it, if not, use SumTools' summation how to simplify it?

`restart;with(QDifferenceEquations):qexp := QPochhammer(-(-p*t), q, infinity);f:= qexp*t;b := subs(t=q^n,f);QSimplify((1-q)*AccurateQSummation(b*q^n,n=-infinity..infinity));with(SumTools):QSimplify((1-q)*Summation(b*q^n,n=-infinity..infinity));`
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