Sparse bounds for discrete singular Radon transforms

We show that discrete singular Radon transforms along a certain class of polynomial mappings $P:{\mathbb{Z}}^d\to {\mathbb{Z}}^n$ satisfy sparse bounds. For $n=d=1$ we can handle all polynomials. In higher dimensions, we pose restrictions on the admissible polynomial mappings stemming from a combination of interacting geometric, analytic and number-theoretic obstacles.