Abstract A linear equation with coefficients in $\mathbb{F}_q$ is common if the number of monochromatic solutions any two-coloring $\mathbb{F}_q^{\,n}$ asymptotically (as $n \to \infty$) at least expected a random two-coloring. The Sidorenko dense subset set same density. In this paper, we characterize those equations which are common, and Sidorenko. main novelty construction based on choosing ...

Given free modules $M\subseteq L$ of finite rank $f\geq 1$ over a principal ideal domain $R$, we give procedure to construct basis $L$ from $M$ assuming the invariant factors or elementary divisors $L/M$ are known. matrix $A\in M_{m,n}(R)$ $r$, its nullspace in $R^n$ is $R$-module $f=n-r$. We submodule $f$ naturally associated with $A$ and whose easily computable, determine quotient module then...