Given a variety $X$, rational equivalence gives an equivalence relation on $Z_i(X)$ leading to the Chow groups and used in the Grothendieck-Riemann-Roch theorem.
I was wondering about replacing $\mathbb{P}^1$ with $\mathbb{A}^1$ in this definition. This would be polynomial equivalence - equivalence via polynomial functions instead of equivalence via rational functions:
Polynomial equivalence: $Z \sim_{pol} Z'$ if there is a cycle $V$ on $X \times \mathbb{A}^1$ flat over $\mathbb{A}^1$, such that $[V \cap X \times \{0 \}] − [V \cap X \times \{ 1 \} ] = [Z] − [Z' ]$.
Is it possible to make sense of $[ ]$ above, and if so does this give an adequate equivalence relation?