Functions and polynomials in vector spaces (Q1092114)

The author shows that a scalar-valued function on a vector space over a finite field, of order q and characteristic p, is polynomial of degree at most n if this is true of its restrictions to all affine subspaces of dimension s where \(s(p-1)q>pn\). This refines the reviewer's results (``Polynomials in modules. I'', to appear in J. Algebra) and is best possible for all p, q, and n. (Over an infinite field, \(s\geq 1\) suffices, reviewer showed.) He also proves a conjecture of the reviewer characterizing n-th degree forms over large prime fields.