@Markiyan Hirnyk 

can quartics relation help to find degree 5s polynomials relation?

how?

@Doug Meade 

actually I would like to learn a general method to understand the properties of variables in a system of polynomial related with right hand side variables.

i would like to research which kind of matrix in right hand side are suitable for a system.

not only this system of polynomials

@vv 

i am doing an experimenr to test whether my system of polynomials generated from polynomial map

so no theoretical facts

@vv 

i have a system of polynomials

and would like to find its polynomial map and 

then find back this system of polynomial to verify this polynomial map belong to it.

but i do not whether this jacobian is your mentioned jacobian,

because it can not convert back to system of polynomials

with(LinearAlgebra):
prej := Matrix([[diff(eq2,a),diff(eq2,b),diff(eq2,c)],[diff(eq3,a),diff(eq3,b),diff(eq3,c)],[diff(eq4,a),diff(eq4,b),diff(eq4,c)]]);
jaco := Determinant(prej);
jaco := -a*b*c^2+c^2;
g3 := [diff(jaco,a),diff(jaco,b),diff(jaco,c)];
K := [r-g3[1],u-g3[2],v-g3[3]];
G := Groebner[Basis](K, 'tord', degrevlex(r,u,v));
R1 := eliminate(G, {r,u,v,w}); # eliminate is the reverse of Basis
Ga := Groebner[Basis]({aa*G[1],aa*G[2],aa*G[3],aa*G[4],aa*G[5],aa*G[6], (1-aa)*K[1], (1-aa)*K[2], (1-aa)*K[3]}, 'tord', deglex(aa,r,u,v));
Ga := remove(has, Ga, [a,b,c,aa]);
K0 := eliminate(Ga, {r,u,v,w});
K0 := eliminate(Ga, {});

@Markiyan Hirnyk 

Jacobian determinant is only one polynomial equation

if a system of 3 polynomials, how can it return polynomial map which has 3 equations?

@Carl Love 

i searched that 

kernel of homomorphism can be ideal

how to do?

@Markiyan Hirnyk 

sorry, i discover above subgroup is not subgroup after

check with membership of subgroup

@Markiyan Hirnyk 

actually i do not know which are independent or dependent variables

can you give at least one example of real practice case or general case?

@Markiyan Hirnyk 

case 1 : all are independent variebles, a,b,c

case 2 : only one independent variable, a

case 3: only one dependent variable a

@Markiyan Hirnyk 

i discover maple 15 and maple 18 's solve has different solutions

actually below i run in maple 18 and expect to convert hilbert series

back to system

how to recover back the system?

sol1 := solve([eq2=1,eq3=0,eq4=s],[a,b,c]);

sol1 := [[a = (-s+1)*RootOf(_Z^2-s^2-_Z+2*s-1)/(RootOf(_Z^2-s^2-_Z+2*s-1)+s^2-2*s+1), b = -(-s+1)*RootOf(_Z^2-s^2-_Z+2*s-1)/(RootOf(_Z^2-s^2-_Z+2*s-1)+s^2-2*s+1), c = RootOf(_Z^2-s^2-_Z+2*s-1)]]

eq2:=s-(-a+1)*c/(c+b^2-2*b+1);
eq3:=s+(-a+1)*c/(c+b^2-2*b+1);
eq4:=c;

solve([eq2=0,eq3=0],s);

a = -sqrt(1/HilbertSeries([eq2, eq3, eq4], {a,b,c}, z))*c/(c+1/HilbertSeries([eq2, eq3, eq4], {a,b,c}, z))

@Christian Wolinski 

rA := {c-a*s+a, s^2-s-a^2*s^2+2*a^2*s-a^2, b*s+a*s-a}, {-s^2+s+r^2}, [RootOf(_Z^2-s^2+s) = r]; X1:=`@`(`union`,op,[proc(L) L[1], map((x->0=x),L[2]) end proc],eliminate)(rA[1],{s}); X2:=`@`(`union`,op,[proc(L) L[1], map((x->0=x),L[2]) end proc],eliminate)(rA[1],{s,b}); X3:=`@`(`union`,op,[proc(E) map(`*`,E,1/b) end proc@select,remove])(has,X2,b);
Error, invalid proc termination

i can not run your code even if add proc to end become end proc

@Christian Wolinski 

sorry, i am wrong in MaxPoly and MinPoly function

i should use integration from negative to positive infinity

since i do not know the range

i guess [-infinity, infinity]  or [0, 1] or [-1,1]

use area to find which is maximum or minimum

moreover, another conjecture of mine is that F(X) in F(x, F(x)) 

may represent an ideal, but i do not know how to substitute ideal into ideal

any one know how to differentiate this in this direction?

@Christian Wolinski 

Hi all,

i use 

solve([eq2=1, eq3=0, eq4=s], [a,b,c]);

to get this solution

this question is asking for eq2, eq3 and eq4

is it possible to find back system in terms of a, b, c ?

@Kitonum 

i rewrite some definitions, but still do not know which definition is correct for this differentiate

any one know how to differentiate this?

restart:
MaxPoly := proc(a,b)
local result, mm1, mm1result, mm2, mm2result:
result := 0:
mm1 := indets(a):
mm1result := subs([seq(mm1[m]=1, m=1..nops(mm1))],a):
mm2 := indets(b):
mm2result := subs([seq(mm2[m]=1, m=1..nops(mm2))],b):
if mm1result >= mm2result then
result := a:
else
result := b:
end if:
return result:
end proc:

MinPoly := proc(a,b)
local result, mm1, mm1result, mm2, mm2result:
result := 0:
mm1 := indets(a):
mm1result := subs([seq(mm1[m]=1, m=1..nops(mm1))],a):
mm2 := indets(b):
mm2result := subs([seq(mm2[m]=1, m=1..nops(mm2))],b):
if mm1result >= mm2result then
result := b:
else
result := a:
end if:
return result:
end proc:

F := (x,y) -> MinPoly(x,y)/MaxPoly(x,y);

so1l := Limit((F(x+h,F(x,y)) - F(x,F(x,y)))/h, h = 0);
sol2 := Limit((F(x,F(x+h,y)) - F(x,F(x,y)))/h, h = 0);
sol3 := Limit((F(x+h,F(x+h,y)) - F(x,F(x,y)))/h, h = 0);
simplify(sol1);
simplify(sol2);
simplify(sol3);

@vv 

i sent this question to technical support of maple with the original code from starting to the end,

since this include my invaluable research equation result, i can not show it here.

hope technical support Matt can reply me