The Linear Artin Approximation Theorem

Let $(R,m,k)$ be a complete local ring. Suppose a system of finitely many equations in $t$ variables over $R$ has a solution $\underline{Z}$ modulo $m^l$. Let $J$ be the ideal generated by the polynomials defining the equations, and assume that

$$J\cap R=(0).$$

Linear Artin Approximation Theorem. Then the system has a true solution $\underline{U}$ such that

$$\underline{Z}-\underline{U}\in m^{\floor{l/c}}R^t,$$

where $c$ is a constant independent of $l$. The authors describe this as a conjecture whose use would simplify their proof of the main result; its resolution status is not established in the supplied text.