with convergence properties better than for Newton method; derivation of a general algorithm for Hilbert tenth problem, linear case, with the extension of a classic Euler theorem from one equation to a system; solvers have been obtained that are more stable than classical ones, especially for the problem arising in primal-dual interior point method; ABS methods are usually faster on vector or parallel machines; ABS methods provide a simpler approach for teaching for a variety of classes of problems, since particular methods are obtained just by specific parameter choices. Knowledge of ABS methods is still quite limited among mathematicians, but they have great potential for improving the methods currently in use. Bibliography Jozsef Abaffy, Emilio Spedicato (1989): ABS Projection Algorithms: Mathematical Techniques for Linear and Nonlinear Algebraic Equations, Ellis Horwood, Chichester. The first monograph on the subject Jozsef Abaffy, Charles G. Broyden, Emilio Spedicato (1984): A class of direct methods for linear equations, Numerische Mathematik 45, 361-376. Paper introducing ABS methods for continuous linear systems. H. Esmaeili, N. Mahdavi-Amiri, Emilio Spedicato: A class of ABS algorithms for Diophantine linear systems, Numerische Mathematik 90''', 101-115. Paper introducing ABS methods for integer linear systems''. Diophantine equations Numerical linear algebra