Variation estimates for averages along primes and polynomials

We prove $q$-variation estimates, $q>2$, on ${\mathrm{\ell }}^p$ spaces for averages along primes (with $1<p<\mathrm{\infty }$) and polynomials (with $\big| \frac1p - \frac12 \big| < \frac{1}{2(d+1)}$,where$d$ is the degree of the polynomial). This improves the pointwise ergodic theorems for these averages in the corresponding ranges of $L^p$ spaces.