Homogeneous additive congruences

Abstract

For prime power $p^n$, let $\mathrm{\Theta }(k,p^n)$ denote the minimal $s$ such that for any integers $a_i$ coprime to $p$ the congruence

$$a_1{x}_1^k+a_2{x}_2^k+\cdots +a_s{x}_s^k\equiv 0(\text{mod}{p}^n)$$

has a primitive solution ($p\nmid x_i$ for some $i$), and ${\mathrm{\Theta }}^{\ast }(k,p^n)$ the minimal $s$ such that there is always a solution with $p\nmid x_i$ for all $i$, $1\le i\le s$. We obtain a number of estimates for these two quantities when the group $A$ of $k$-th powers mod $p$ has at least two elements, including, $\mathrm{\Theta }(k,p)\ll {\text{log}}^{3}k$, $\mathrm{\Theta }(k,p)\ll {\text{log}}^{5∕2}k$ if $\left|A\right|$ is composite, $\mathrm{\Theta }(k,p^n)\ll k^{\sqrt{5}∕\sqrt{{\text{ log}}_{2}k}}{\text{ log}}^{2}k,$ ${\mathrm{\Theta }}^{\ast }(k,p^n)\le 8k^{5∕{\text{log }}_2(t_1∕C)}$ for some absolute constant $C$, where , and ${\mathrm{\Theta }}^{\ast }(k,p)\le 8$ for .