Completeness of the ring of polynomials

Consider the polynomial ring R:=k[X₁,…,X_n]R:=k[X1,…,Xn] in n≥2n≥2 variables over an uncountable field k. We prove that R   is complete in its adic topology, that is, the translation invariant topology in which the non-zero ideals form a fundamental system of neighborhoods of 0. In addition we prove that the localization R_mRm at a maximal ideal m⊂Rm⊂R is adically complete. The first result settles an old conjecture of C.U. Jensen, the second a conjecture of L. Gruson. Our proofs are based on a result of Gruson stating (in two variables) that R_mRm is adically complete when R=k[X₁,X₂]R=k[X1,X2] and m=(X₁,X₂)m=(X1,X2).