ON A THEOREM OF AYAD AND RYCKELYNCK

Abstract

Let d be a K-derivation of the polynomial ring K[x ₁ , …, x _n ] over a field K of characteristic 0, and let [dtilde] 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 [dtilde] is equal to the fraction field Q(Ker d) of Ker d. In this note, we give a short proof for this result.

Key Words:

• Derivation
• Polynomial ring

Mathematics Subject Classification:

• Primary 13N15
• Secondary 13F20

ACKNOWLEDGMENT

I would like to express my gratitude to the referee for his helpful suggestions and valuable comments.

Notes

^‡Dedicated to Professor Motoshi Hongan on his 60th birthday.

^#Communicated by M. Ferrero.