In this paper we show that any system of equations over a free nilpotent group of class c is either unitary or nullary. In fact, such a system either has a most general solution (akin to the most general solution of a system of linear diophantine equations), or every solution has a proper generalization. In principle we provide an algorithm for determining whether or not a most general solution...

Linear algebraic techniques based on minimization of thermodynamic functional and/or other constraints are illustrated for mass balance of chemical reactions that do not exhibit stoichiometrically unique solutions in the linear algebraic vector space. The techniques demonstrate elegant casting of chemical equations in terms of generalized linear Diophantine matrices and generalized elimination ...

We develop an algorithm for solving a diophantine equation with lower and upper bounds on the variables. The algorithm is based on lattice basis reduction, and rst nds short vectors satisfying the diophantine equation. The next step is to branch on linear combinations of these vectors, which either yields a vector that satis es the bound constraints or provides a proof that no such vector exist...

A “diophantine equation” is a system of polynomial equations where we restrict the solution set to rational or integral values. The study of diophantine equations is the study of the nature of these integral or rational solution sets. “diophantine geometry” is the study of diophantine equations using the language of algebraic geometry. Specifically, one associates to a diophantine equation the ...