Symbol length of p-algebras of prime exponent

We prove that if the maximal dimension of an anisotropic homogeneous polynomial form of prime degree p over a field F with char( F) - p is a finite integer d greater than 1 then the symbol length of p-algebras of exponent p over F is bounded from above by [d-1/p] - 1 and show that every two tensor products of symbol algebras of lengths k and l with ( k + l)p >= d - 1 can be modified so that they share a common slot. For p = 2, we obtain an upper bound of u(F)/2 - 1 for the symbol length, which is sharp when I-q(3) F = 0.