Polynomial progressions in topological fields

Let $P_1,\dots ,P_m\in K[y]$ be polynomials with distinct degrees, no constant terms and coefficients in a general locally compact topological field $K$. We give a quantitative count of the number of polynomial progressions $x,x+P_1(y),\dots ,x+P_m(y)$ lying in a set $S\subseteq K$ of positive density. The proof relies on a general $L^{\mathrm{\infty }}$ inverse theorem which is of independent interest. This inverse theorem implies a Sobolev improving estimate for multilinear polynomial averaging operators which in turn implies our quantitative estimate for polynomial progressions. This general Sobolev inequality has the potential to be applied in a number of problems in real, complex and $p$-adic analysis.