Let L_i(X), i=1,2, L₁≠ L‾₂, be two complex linear forms in R⁴, and K_i(X)=L_i(X)L‾_i(X) are positive quadratic forms. The root sets L_i of forms K_i are two-dimensional planes in R⁴. Assume that L₁∩L₂=0 and that there are no integer points except 0 which lie at L_i. We propose an algorithm of computation of integer points that give the best approximations to the sets of roots L_i. If coefficients of forms L_i lie in totally complex quaternary conjugated number fields k_i, our algorithm often finds unities of k_i. The algorithm was tested on the set of quaternary number fields specified by equations with small coefficients. The algorithm was successful more often than the best of known algorithms in totally real quaternary case - the Güting algorithm.

Mathematical problems and theory of numerical methods