On a theorem of Ayad and Ryckelynck

摘要

Let d be a K-derivation of the polynomial ring K[x(1),..., x(n)] over a field K of characteristic 0, and let d be the extension of d to the fraction field K(x(1),..., x(n)). Recently Ayad and Ryckelynck proved that if the kernel Ker d of d contains n - 1 algebraically independent polynomials then Ker is equal to the fraction field Q(Ker d) of Ker d. In this note, we give a short proof for this result.