Divisibility of exponential sums via elementary methods
Let \(E=(e_{ij})\) be an \(n\times N\) non-negative integer matrix whose columns \({\mathbf e}_j=(e_{ij})_{i=1}^n\) are distinct and whose rows \(R_i=(e_{ij})_{j=1}^N\) are non-zero. The zero-rank \(r_E(\nu)\) of a non-negative integer \(N\)-tuple \(\nu=(\nu_1,\ldots,\nu_N)\) is the number of rows \(R_i\) of \(E\) such that \(R_i\cdot\nu=0\). Call \(\nu\) an \(m\)-covering when \(E\nu^T=\nu_1{\mathbf e}_1+\cdots+\nu_N{\mathbf e}_N\) has all its entries non-zero and divisible by \(m\). The \(m\)-th covering number \(\kappa_m(E)\) of \(E\) is the least value of \(\nu_1+\cdots+\nu_N\) for such an \(m\)-covering. Let \(F\) be a polynomial in \({\mathbb F}_p[{\mathbf x}]\) where \({\mathbf x}=(x_1,\ldots,x_n)\) and set \(S(F)=\sum_{{\mathbf x}\;\text{in}\;S^n} \zeta^{F({\mathbf x})}\) where \(\zeta\) is a primitive \(p\)-th root of unity and \(S=\{0,1,\ldots,p-1\}\). Denote by \(\nu_p\) the extension to \({\mathbb Q}(\zeta)\) of the \(p\)-adic valuation. Define \({\mathbb F}_p[{\mathbf x}]_E =\{\sum_{j=1}^N a_jx_1^{e_{1j}} \cdots x_n^{e_{nj}} :a_1,\ldots,a_N\;\text{in}\;{\mathbb F}_p\}\), a vector subspace of \({\mathbb F}_p[{\mathbf x}]\) of dimension \(N\). The authors show that \(\nu_p(S(F))\geq \kappa_{p-1}(E)/(p-1)\) and there is an \(F\) in \({\mathbb F}_p[{\mathbf x}]_E\) for which equality holds. This is proved by \(\theta\)-adic analysis where \(\theta=1-\zeta\) is the prime ideal in \({\mathbb Z}[\zeta]\) with \((p)=(\theta)^{p-1}\). Let \(F(x_1,\ldots,x_n)=\sum_{j=1}^N a_jx_1^{e_{1j}}\cdots x_n^{e_{nj}}\). Let \(\Delta(F)\) be the Newton polyhedron of \(F\), that is the convex hull in \({\mathbb R}^n\) of the set \(\{{\mathbf e}_j\}\cup\{0,\ldots,0\}\). Let \(\omega(F)\) be the smallest positive rational number such that \(\omega(F)\Delta(F)\) contains a point of \({\mathbb Z}^n\) with positive coordinates. The authors show that \(\kappa_{p-1}(E)/(p-1)\geq \omega(F)\). This leads to the theorem: Let \(P_1({\mathbf x}),\ldots,P_t({\mathbf x})\) in \({\mathbb F}_p[x_1,\ldots,x_n]\) and let \(N\) be the number of common zeros of \(P_1,\ldots,P_t\). Introduce \(t\) extra variables \(y_1,\ldots, y_t\) and define a new polynomial \(F\) in \(n+t\) variables by \(F({\mathbf x}, {\mathbf y})=y_1P_1({\mathbf x})+\cdots+y_tP_t({\mathbf x})\). Then \(p^{\kappa_{p-1}(F)/(p-1)-t}\) divides \(N\) and the divisibility is tight. The approach gives an elementary proof of the prime field case of the result of [\textit{O. Moreno, K. Shum, F. N. Castro} and \textit{P. J. Kummer}, ''Tight bounds for Chevalley-Warning-Ax type estimates with improved applications.'' Proc. Lond. Math. Soc. (3) 88, No. 3, 545--564 (2004; Zbl 1102.11032)].
scientific article; zbMATH DE number 5757998