On Banach Spaces Whose Unit Sphere Determines Polynomials

In this paper we study the problem of characterizing the real Banach spaces whose unit sphere determines polynomials, i.e., if two polynomials coincide in the unit sphere, is this sufficient to guarantee that they are identical? We show that, in the frame of spaces with unconditional basis, nonre flexivity is a sufficient, although not necessary, condition for the above question to have an affirmative answer. We prove that the onlylnpspaces having this property are those with p irrational, while the only l p spaces which do not enjoy it are those with p an even integer. We also introduce a class of polynomial determining sets in any real Banach space.