update1 i notice solving 1+x+x^2+x^3+x^4+x^5 have 5 solutions two conjugate real number and each of them having conjugate complex number part
i change
a*b + c
to
a1*a2*b1*b2 + c1*c2**
still can not solve non-isomorphic problem in degree 5 or more
is it the same problem meet by lagrange
old question when design system of polynomials of degree 5, after solve it, it return two or more set of roots
how to do isomorphic ideal from root system of degree 5 or more
solve({f1=s,f2=s,f3=s}) {[a=1,b=2,c=s] , [a=3,b=4,c=s]} there are two possible set of roots
first set is a=1, b=2,c=s second set is a=3, b=4,c=s
which method can ensure only one set of root of system of polynomials when design?
Let's suppose you want to design a system of three polynomials $P_1$, $P_2$, $P_3$ in three variables $x,y,z$ with only one real solution: for convenience take that solution to be $x=0, y=0, z=0$. The zeros of each polynomial will in general form a variety $Z(P_i)$, typically a surface, in $\mathbb R^3$, and you want $(0,0,0)$ to be the only point where all three varieties intersect. There's lots of freedom to choose the polynomials, as long as you keep that in mind. If the varieties for the first two polynomials are surfaces, they will typically intersect on a curve. You then just have to make sure the third variety doesn't intersect that curve except at $(0,0,0)$.
One way that will work is to make $$P_3(x,y,z) = f(x,y,z) P_1(x,y,z) + g(x,y,z) P_2(x,y,z) + x^2 + y^2 + z^2$$ for some polynomials $f$ and $g$.