Symbol Length of $p$-Algebras of Prime Exponent

Abstract: We prove that if the maximal dimension of an anisotropic homogeneous polynomial form of prime degree $p$ over a field $F$ with $\operatorname{char}(F)=p$ 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 $\left \lceil \frac{d-1}{p} \right \rceil-1$, and show that every two tensor products of symbol algebras of lengths $k$ and $\ell$ with $(k+\ell) p \geq d-1$ can be modified so that they share a common slot. For $p=2$, we obtain an upper bound of $\frac{u(F)}{2}-1$ for the symbol length, which is sharp when $I_q^3 F=0$.