  We address the problem of unambiguous reconstruction of rational functions of
many variables. This is particularly relevant for recovery of exact expansion
coefficients in integration-by-parts identites (IBPs) based on modular
arithmetic. These IBPs are indispensable in modern approaches to evaluation of
multiloop Feynman integrals by means of differential equations. Modular
arithmetic is far more superior to algebraic implementations when one deals
with high-multiplicity situations involving a large number of Lorentz
invariants. We introduce a new method based on balanced relations which allows
one to achieve the goal of a robust functional restoration with minimal data
input. The technique is implemented as a Mathematica package Reconstruction.m
in the FIRE6 environment and thus successfully demonstrates a proof of concept.
