${\mathrm{\ell }}^p({\mathbb{Z}}^n)$-estimate for long $r$-variational seminorm of discrete Birch-Magyar averages

We prove ${\mathrm{\ell }}^p({\mathbb{Z}}^n)-$estimates for long $r$-variational seminorm of two families of averages: discrete Birch-Magyar averages, for $r>max\{p,p^{\mathrm{\prime }}\}$ with $p>\frac{2c_{\mathfrak{R}}-2}{2c_{\mathfrak{R}}-3}$ and discrete Hardy-Littlewood type averages over certain algebraic varieties, for $r>max\{p,p^{\mathrm{\prime }}\}$ with $p>1$. Further, we discuss an application of these results in ergodic theory.