Thakur's hypotheses on power sums of $\mathbb{F}_q[t]$

Abstract

In his 2009 paper, Thakur posed three conjectural hypotheses for the degrees of the power sums \[ S_d(k)=\sum_{\substack{a\in \mathbb F_q[t] \text{ monic}\\ °a=d}} a^{-k} \qquad\text{and}\qquad s_d(k)=-°_t S_d(k). \] For prime fields $q=p$, we prove Hypotheses H1 and H2, giving a unique greedy description of the extremal term in Carlitz&#39;s formula and establishing the recursion \[ s_d(k)=s_{d-1}(s_1(k))+s_1(k). \] As consequences, the prime-field recursion gives the strict Newton-polygon convexity used in the prime-field Carlitz-Goss Riemann-hypothesis theorem, and it recovers Thakur&#39;s nonvanishing theorem for positive multizeta values over $\mathbb F_p[t]$. We also prove Hypothesis H3 for all finite fields $q=p^f$, establishing the monotonicity \[ s_d(k)<s_d(k+1)\qquad (p\nmid k). \] We provide Lean formalizations of the arguments in this paper, generated by AxiomProver.