Waring's problem for algebras over fields (Q1092106)

For a ring A let \(w_ k(A)\) denote the smallest s such that every sum of kth powers in A is a sum of s kth powers; let \(v_ k(A)\) denote the corresponding infimum for the sum-or-difference of kth powers. This paper gives a detailed study of \(v_ k(A)\) and \(w_ k(A)\) when A is an algebra over a field F. The main result can be stated as follows: \(v_ k(A)\leq k^ 3\) when A is commutative and either F is infinite or A has transcendence degree 1 over F. This statement swallows up several more precise results, which give sharper upper bounds in many situations of interest.