About 2 decades ago I heared from some one that there are infinitely many open contractible subsets of space mutually non homeomorphic to each other. I confess that I do not remember the details of construction but I remember the construction was not so routine(If I am not mistaken the method was based on infinitness of the set of prime numbers..)

But can such examples be constructed in an algebraic manner? In particular are there two polynomials $H,G: \mathbb{R}^3 \to \mathbb{R}$ and two open sets $U,V$ in $\mathbb{R}$ such that $H^{-1}(U)$ and $G^{-1}(V)$ are non homeomorphic contractible sets?