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Let k be a field,and let R be an integral k-domain.R has the polynomial cancellation property if,for any integral k-domain S and n ≥ 0,the condition R[n](≈)k S[n] implies R(≈)=k S.Similarly,R has the Laurent cancellation property if R[±n](≈)k S[±n] implies R(≈)=k S.Suppose that R has transcendence degree one over k.The 1972 theorem of Abhyankar-Eakin-Heinzer shows that R has the polynomial cancellation property.We give a concise new proof of this result(in the characteristic zero case)using the Makar-Limanov invariant of R,which is the ring of invariants of all k+-actions on R.Our proof features an affine version of a theorem for fields due to Nagata,which is of independent interest.Next,we will prove that R also has the Laurent cancellation property.This proof uses the neutral subring of R,which we define to be the ring of invariants of all k*-actions on R.One goal of the talk is to highlight the similar role played by the Makar-Limanov invariant and the neutral subring in these two proofs.