i'm looking for an inverse function in a 3 dimensions space :
$f~:~[0,1]^3\to[0,1]^3$
$$f(x,y,z)=\begin{pmatrix}x(1-(y+z)/2+yz/3)\\y(1-(x+z)/2+xz/3)\\z(1-(y+x)/2+yx/3)\end{pmatrix}$$ Does anybody know the easiest way to start this problem?
Thank you.
This problem was solved by using Groebner Basis.
If $(a,b,c)=f(x,y,z)$ then:
It is equivalent to solving:
$$ x^6 + (-5a + b + c - 7)x^5 + (4a^2 - 4ab - 4ac + 4bc + 35a - 7b - 7c + 19)x^4 + (-28a^2 + 22ab - 2b^2 + 22ac - 12bc - 2c^2 - 92a + 16b + 16c - 24)x^3 + (73a^2 - 40ab + 3b^2 - 40ac + 10bc + 3c^2 + 108a - 12b - 12c + 12)x^2 + (-84a^2 + 24ab + 24ac - 48a)x + 36a^2=0$$
then
$$(x^3 - 3x^2 + 2x)y - 2x^3 + 2x^2a - 2x^2c + 7x^2 - 7xa + xb + 3xc - 6x + 6a=0$$
and finally:
$$z + yx^2 - yx - 2x^2 + 2xa - 2xc - y + 3x - 3a + b - c=0$$