Efficient congruencing in $\mathbb Z[t]$ modulo $t^2$

Abstract

We develop an analogue of nested efficient congruencing applicable in the ring $\mathbb Z[t]/t^2\mathbb Z[t]$ so as to estimate mean values related to that of Vinogradov. Let $k$ be a natural number, and suppose that $φ_j(u)\in \mathbb Z[t,u]$ $(1\le j\le k)$ is a system of polynomials having Wronskian $W(u;\boldsymbol φ)=\det ({\rm d}^iφ_j(u)/{\rm d}u^i)_{1\le i,j\le k}$ with respect to $u$. Consider the number $Υ_{s,k}(X;\boldsymbol φ)$ of integral solutions of the system of equations \[ \sum_{i=1}^s \bigl( φ_j(u_i+v_it)-φ_j(u_{s+i}+v_{s+i}t)\bigr) \equiv 0 \pmod{t^2}\quad (1\le j\le k), \] with $|u_i|\le X$ and $|v_i|\le X$ $(1\le i\le s)$. We show that when $W(u;\boldsymbol φ)\not\equiv 0\pmod{t}$ and $1\le s\le k(k+1)/2$, then $Υ_{s,k}(X;\boldsymbol φ)\ll X^{2s+\varepsilon}$. Our conclusions extend to number fields and function fields, as we outline. We describe several consequences of our results, including forthcoming work establishing the main conjecture for certain systems of equations not possessing translation-dilation invariant properties.