Multivariable Congruence Jumping: Periodic States, Integral Realizations, and Descent

Abstract

Congruence jumping replaces coordinates by integer quotients and updates polynomial divisibility relations. We extend this procedure to several variables and retain integer coefficients by recording their common factor separately. For three-variable systems of divisibilities with signed binomial numerators, we classify all positive integer exponent tuples of period at most two under three successive coordinate jumps: seven fixed tuples and twenty-two cycles of exact period two. No odd period greater than one occurs. At the chosen representatives of the twenty-two exponent two-cycles, there are eighty-four returning sign assignments. All have realizations with nonzero pairwise-coprime integer coordinates, and exactly seventy-four have such realizations with every coordinate positive throughout the six jumps. Explicit constructions and congruence arguments establish this classification. We also study jumps that decrease a specified height, distinguishing points that admit no immediate decrease from minima of entire connected components of the jump graph. For the four-variable family $Q_λ(x,y,z,w)=kxyzw$, where $Q_λ$ is the sum of the four squares plus $λ$ times the six pairwise products, finite bounds give complete classifications for $λ=1,2,3$. For $λ=1$, positive integer solutions exist exactly for $k=2,8,10$. These results distinguish returns of the polynomial data from integrality and positivity of the coordinate sequences.