Items tagged with groebner

eq1a := Homogenize(eq1, h);
eq2a := Homogenize(eq2, h);
eq3a := Homogenize(eq3, h);
David Cox using Algebraic Geometry page 82 use resultant to eliminate variable h
eq1b := eq1a - x;
eq2b := eq2a - y;
eq3b := eq3a - z;
GB := Basis([eq1b,eq2b,eq3b], T2);
r1 := resultant(eq1b, eq2b, h);
r2 := resultant(eq1b, eq3b, h);
r1 = r2

page 82 teach how to eliminate, after do question 2, discover r1 and r2 are the same.

how to eliminate the variable h with resultant after homogenize ideal with variable h

is it possible to find the input ideal from given hilbert series ?

i ask this because maple out of memory when computing all combination of ideals

which book teach how to design hilbert series

i want to research hilbert series and construct similar invariants to classify ideals

i do not know how many memory in order to do this. hope maple can display how much memory need before computation of all combination and display how long will wait for it finish computation.

my home computer can not afford this computation


Let f:X->f(X)  be a polynomial function from C^n to C^p. Let r(X) be the rank of the Jacobian matrix of f in X. What is the maximal value of r(X) when X goes throught C^n ?

In other words, I'd want to obtain the maximal dimension of the components of im(f). 

How to proceed ?

Thanks in advance.

I try to sort a polynomial using the graded reverse lexicograpic order. According to the Documentation this is achieved via tdeg.
So here is what i tried:


sort( x+y+z, order = tdeg(x,y,z));


sort(x+y+z, [x,y,z], tdeg);

In both cases maple returns "x+y+z" instead of the expected "z+y+x". What am i doing wrong?



computer a Gröbner basis for <f_[1] = x^2*y - 2*y*z + 1, f_[2] = x*y^2-z^2+ 2*x,  f_[3] = y^2*z - x^2+ 5 > belong to Q[x,y,z], using ≺= <_grlex with x≺y≺z. compare your output to the Gröbner basis the Maple computers with a different order.

T := lexdeg([x1,x2,x3],[e1,e2,e3]);
GB := Basis([e1+.999987406876435, e2-.999919848203811],T):

Error, (in LinearAlgebra:-Basis) invalid input: LinearAlgebra:-Basis expects its 1st argument, V, to be of type {Vector, {set(Vector), list(Vector)}} but received [e1+HFloat(0.9999874068764352), e2-HFloat(0.9999198482038109)]

T := lexdeg([x1,x2,x3,x4,x5,x6,x7,x8],[e1,e2]);
GB := Basis([e1-x4*x2^3*x5/(x3*x1^4*x6), e2-x1*x8/(x3*x7)],T):

Error, (in LinearAlgebra:-Basis) invalid input: LinearAlgebra:-Basis expects its 1st argument, V, to be of type {Vector, {set(Vector), list(Vector)}} but received [e1-x4*x2^3*x5/(x3*x1^4*x6), e2-x1*x8/(x3*x7)]

T := lexdeg([x1,x2,x3],[e1,e2,e3]);
hello1 := proc(xx,yy)
return MatrixMatrixMultiply(xx,yy);
end proc:
hello2 := proc(xx,yy)
return xx+yy- MatrixMatrixMultiply(xx,yy);
end proc:
m1 := Matrix(3, 3, {(1, 1) = -.737663975994461+0.*I, (1, 2) = -.588973463383001+0.*I, (1, 3) = .330094104689369+0.*I, (2, 1) = -.588012653178741+0.*I, (2, 2) = .320157823261769+0.*I, (2, 3) = -.742792089286083+0.*I, (3, 1) = -.331802619371428+0.*I, (3, 2) = .742030476217061+0.*I, (3, 3) = .582492741708719+0.*I});
m2 := Matrix(3, 3, {(1, 1) = -.742269137704830+0.*I, (1, 2) = -.590598631673326+0.*I, (1, 3) = .316590877121441+0.*I, (2, 1) = -.593533033362923+0.*I, (2, 2) = .360143915024171+0.*I, (2, 3) = -.719732518911068+0.*I, (3, 1) = -.311054762892221+0.*I, (3, 2) = .722142379823161+0.*I, (3, 3) = .617863510611693+0.*I});
m3 := Matrix(3, 3, {(1, 1) = -.751491355856820+0.*I, (1, 2) = -.574908634018322+0.*I, (1, 3) = .323636840615627+0.*I, (2, 1) = -.575794245520782+0.*I, (2, 2) = .332066412772496+0.*I, (2, 3) = -.747123071744916+0.*I, (3, 1) = -.322058579916187+0.*I, (3, 2) = .747804760642505+0.*I, (3, 3) = .580574121936877+0.*I});
AA := hello1(m1, m2);
BB := hello2(m1, m2);
GB := Basis([e1- AA,e2- BB],T):
NormalForm(m3, GB, T);

Two questions:

The algortihms that Groebner[Basis] uses at each step computes some "tentative" or "pseudo-basis". The "tentative" basis is not a Groebner basis but it is in the ideal generated by the original system of polynomial eq.

1) Is this correct ? Provided this is correct, then

2) How can one retrive the last "tentative" basis?
 If I just use timelimit I can abort the computations but how can one retrive the last computation?


I have been struggling (reading Ore/Weyl Algebra documentation) to understand how to input a PDE system with polynomial coeff. in Weyl algebra notation so I can compute a Groebner basis for it. I would be very grateful if someone could  show, using the simple example below, which differential operators in Ore_algebra[diff_algebra] should one declare to express the system in Weyl algebra notation. The systems I'm working are more complicated but all have many dependent variables, f and g functions in this example:

pdesys:= [ x*diff( f(x,y,z),x)- z*diff( g(x,y,z),y) = 0, (x^2-y)*diff( f(x,y,z),z)- y*diff( g(x,y,z),z) = 0 ]

DoExist := proc(tau, n)

if rtable_num_elems(tau) >= n then

        return tau[n];


        return 0;

end if;

end proc;


g1 := [0, y, x];

g2 := [0, y^2-x-y, 0];

g3 := [x, x+y, 0];

g4 := [y, -y, 0];

g5 := [0, x*y+x/2+y/2, 0];

g6 := [0, x^2-x/4-y/4, 0];


m1 := Basis(g1, tdeg...

g1 := Vector([1, 1]);g2 := Vector([x+y, 0]);g3 := Vector([y^2+1, 0]);g4 := Vector([0, y^3+y]);g5 := Vector([0, x-y]);

i pass and tdeg(x, y, h251, h252, h253, h254, h255) to Basis

failed in Maple 15

above link's comment said eliminate will NOT generate a polynomial in all cases

make me think that eliminate in above link can not return correct one.

what is the correct way to convert groebner basis of kernel into kernel?

the goal is to check kernel belong to image in Maple

K := {r-x^4,u-(x^3)*y,v-x*y^3,w-y^4};
G := Basis(K, 'tord', degrevlex(r,u,v,w));
R1 := eliminate(G, {r,u,v,w}); # eliminate is the reverse of Basis
Ga := Basis({a*G[1],a*G[2],a*G[3],a*G[4],a*G[5],a*G[6],a*G[7],a*G[8],a*G[9],a*G[10],a*G[11],a*G[12],a*G[13],a*G[14], (1-a)*K[1], (1-a)*K[2], (1-a)*K[3], (1-a)*K[4]}, 'tord', deglex(a,r,u,v,w));
Ga := remove(has, Ga, [x,y,a]);

below code is calculate basis of kernel and kernel

i guess basis of image is 

remove(has, Ga, [r,u,v,w]); if this correct, i eliminate this, i can get the image
however it include variable 'a'
is it correct? if not, how to calculate? 
my final goal is to make unexact sequence into exact sequence
1 2 3 4 Page 3 of 4