Essential dimension of symmetric groups in prime characteristic

The essential dimension $\operatorname{ed}_k({\rm S}_n)$ of the symmetric group ${\rm S}_n$ is the minimal integer $d$ such that the general polynomial $x^n + a_1 x^{n-1} + \ldots + a_n$ can be reduced to a $d$-parameter form by a Tschirnhaus transformation. Finding this number is a long-standing open problem, originating in the work of Felix Klein, long before essential dimension was formally defined. We now know that $\operatorname{ed}_k({\rm S}_n)$ lies between $\lfloor n/2 \rfloor$ and $n-3$ for every $n \geqslant 5$ and every field $k$ of characteristic different from $2$. Moreover, if $\operatorname{char}(k) = 0$, then $\operatorname{ed}_k({\rm S}_n) \geqslant \lfloor (n+1)/2 \rfloor$ for any $n \geqslant 6$. The value of $\operatorname{ed}_k({\rm S}_n)$ is not known for any $n \geqslant 8$ and any field $k$, though it is widely believed that $\operatorname{ed}_k({\rm S}_n)$ should be $n-3$ for every $n \geqslant 5$, at least in characteristic $0$. In this paper we show that for every odd prime $p$ there are infinitely many positive integers $n$ such that $\operatorname{ed}_{\mathbb F_p}(\rm{S}_n) \leqslant n-4$.