On Serre's problem on projective modules

The main purpose of this paper is to prove the following result concerning Serre's problem: Any projective module of rank $n$ over $k[{X_1}, \cdots ,{X_n}]$ (where $k$ is an infinite field) is free. We give also simple proofs (based on Serre's theorem that ${K_0}(k[{X_1}, \cdots ,{X_n}]) = Z)$ to the following particular case of Bass' theorem: any projective module of rank $> n$ over $k[{X_1}, \cdots ,{X_n}]$ ($k$ any field) is free, and to Seshadri's theorem: finitely generated projective modules over $k[X,Y]$ are free.