Multidimensional Vinogradov-type estimates in function fields (Q2876531)

A technical paper. Method of proof: a variant of the circle method using Wooley's efficient congruence method. In order to give one of the three main results we need some notation: Let \(q\) be a power of a prime number \(p.\) For a given non-negative integer \(N\) we put \(I_N := \{A \in \mathbb{F}_q[t] : \deg(A) < N\}\). Let \(R\) be a finite subset of \(\mathbb{N}^d\) such that for each \(j := (j_1,\ldots,j_d) \in R,\) if \(k := (k_1,\ldots,k_d) \in \mathbb{N}^d\) satisfies \({j_1 \choose k_1} \cdots {j_d \choose k_d} \not\equiv 0 \pmod p\), then \(k \in R.\) Let \(J_{s}(R;N) := \mathrm{card}(\{u_1^j + \cdots + u_s^j = v_1^j + \cdots + v_s^j ), j \in R, u_j,v_j \in I_N^d \}).\) Finally, let \(R_1 := \{ j \in \mathbb{N}^d : j \not\equiv 0 \pmod p,\;\text{and}\;p^a j \in R,\;\text{for some}\;a \in \mathbb{N}\}.\)NEWLINENEWLINE Theorem. Let \(r := \mathrm{card}(R_1)\), \(\phi := \max_{j \in R_1} | j |,\) and \( k := \sum_{j \in R_1} | j|.\) If \(d \geq 2,\) \(\phi \geq 2\) and \(s \geq r(\phi+1)\) then for each \(\varepsilon >0\), there exists a positive constant \(C\) depending on \(s,d,r,\phi,k,q,\varepsilon\) such that NEWLINE\[NEWLINE J_s(R;N) \leq C q^{N(2s d - k + \varepsilon)}. NEWLINE\]