$$p$$-Adic Polynomial Regression Detecting Digitwise Noise

We propose a new formulation of $$p$$ -adic optimisation as the infinitesimal limit of the least squares method, and introduce several algorithms for $$p$$ -adic optimisation. Since the optimisation problem includes the maximal feasible subsystem problem of linear equations over the finite field $$\mathbb{F}_p$$ , which is APX-complete, i.e. complete for the class of problems which allow constant-factor approximations, by E. Amaldi and V. Kann, we mainly deal with heuristic approaches to the $$p$$ -adic optimisation under mild assumptions. In particular, we deal with $$p$$ -adic polynomial regression under the assumption that noise occurs digitwise sparsely.