Cancellation problems and dimension theory

A well-known cancellation problem of Zariski asks - for two given domains (fields, respectively) K 1 and K 2 over a field k - whether the k -isomorphism of K₁[t] (K(t), respectively) and K₂[t] (K₂(t), respectively) implies the k -isomorphism of K₁ and K₂. In this article, we address and systematically study a related problem: whether the k-embedding from K₁[t](K₁(t), respectively) into K₂[t](K₂(t), respectively) implies the k-embedding from K 1 into K₂. Our results are affirmative: if K₁ and K ₂ are affine domains over an arbitrary field k, and K₁[t] can be k -embedded into K₂ [t], then K 1 can be k -embedded into K₂; if K₁ and K₂ are affine fields over an arbitrary field k, and K₁ t) can be k-embedded into K₂ (t), then K₁ can be k-embedded into K₂. Similar results are obtained for some general nonaffine domains and nonaffine fields. These results were obtained in Belov et al. (preprint) together with L. Makar-Limanov. In this article we give an alternative proof, show connection with dimension theory, consider the case of infinite transcendental degree, and present some applications and surroundings.