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 \(\operatorname{ed}_{\mathbb F_p}(\rm{S}_n) \leqslant n-4\).