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International Conference on Rewriting Techniques and Applications

RTA 1997: Rewriting Techniques and Applications pp 269–283Cite as

Solving linear Diophantine equations using the geometric structure of the solution space

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1232))

Abstract

In the development of algorithms for finding the minimal solutions of systems of linear Diophantine equations, little use has been made (to our knowledge) of the results by Stanley using the geometric properties of the solution space. Building upon these results, we present a new algorithm, and we suggest the use of geometric properties of the solution space in finding bounds for searching solutions and in having a qualitative evaluation of the difficulty in solving a given system.

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References

  1. Boudet, A., Contejean E., and Devie, H.: A new AC Unification algorithm with an algorithm for solving systems of Diophantine equations. In Proceedings of the 5th Conference on Logic and Computer Science, IEEE, 289–299, 1990.

    Google Scholar 

  2. Clausen, M., and Fortenbacher, A.: Efficient solution of linear Diophantine equations. J. Symbolic Computation, 8, 201–216, 1989.

    Google Scholar 

  3. Domenjoud, E.: Outils pour la Déduction Automatique dans les Théories Associatives-Commutatives. Thése de doctorat, Université de Nancy I, 1991.

    Google Scholar 

  4. Elliott, E. B.: On linear homogenous Diophantine equations. Quart. J. Pure Appl. Math., 34, 348–377, 1903.

    Google Scholar 

  5. Filgueiras, M., and Tomás, A. P.: Fast Methods for Solving Linear Diophantine Equations. In M. Filgueiras, L. Damas (eds.) Progress in Artificial Intelligence — 6th Portuguese Conference on Artificial Intelligence, Lecture Notes in Artificial Intelligence 727, Springer-Verlag, 297–306, 1993.

    Google Scholar 

  6. Filgueiras, M., and Tomás, A. P.: A Fast Method for Finding the Basis of Non-negative Solutions to a Linear Diophantine Equation. J. Symbolic Computation, 19, 507–526, 1995.

    Google Scholar 

  7. Huet, G.: An algorithm to generate the basis of solutions to homogeneous linear Diophantine equations. Information Processing Letters, 7(3), 1978.

    Google Scholar 

  8. Lambert, J.-L.: Une borne pour les générateurs des solutions entières positives d'une équation diophantienne linéaire. Comptes Rendus de l'Académie des Sciences de Paris, t. 305, série I, 39–40, 1987.

    Google Scholar 

  9. MacMahon, P.: Combinatory Analysis, 2. Chelsea Publishing Co., 1918.

    Google Scholar 

  10. Moulinet-Ossola, C: Algorithmique des Réseaux et des Systémes Diophantiens Linéaires. Thèse de doctorat, Université de Nice Sophia-Antipolis, 1995.

    Google Scholar 

  11. Petitjean, E.: Résolution Parallèle de Contraintes Linéaires sur les Entiers Naturels. Mémoire de DEA, Université de Nancy I, 9/1996.

    Google Scholar 

  12. Pottier, L.: Minimal solutions of linear diophantine systems: bounds and algorithms. In R. V. Book (ed.), Proceedings of the 4th International Conference on Rewriting Techniques and Applications, Lecture Notes in Computer Science 488, Springer-Verlag, 162–173, 1991.

    Google Scholar 

  13. A. Schrijver, Theory of Linear and Integer Programming, Wiley-Interscience, 1986.

    Google Scholar 

  14. Stanley, R.P.: Linear homogeneous Diophantine equations and magic labelings of graphs. Duke Math. J., 40, 607–632, 1973.

    Google Scholar 

  15. Stanley, R.P.: Enumerative Combinatorics, Vol I, The Wadsworth & Brooks/Cole Mathematics Series, 1986.

    Google Scholar 

  16. Tomás, A. P. and Filgueiras, M.: A new method for solving linear constraints on the natural numbers. In P. Barahona, L. Moniz Pereira, A. Porto (eds.), Proceedings of the 5th Portuguese Conference on Artificial Intelligence, Lecture Notes in Artificial Intelligence 541, Springer-Verlag, 30–44, 1991.

    Google Scholar 

  17. Tomás, A. P.: On Solving Linear Diophantine Constraints. Tese de Doutoramento, submitted to Faculdade de Ciências da Universidade do Porto, 1997.

    Google Scholar 

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Authors and Affiliations

  1. LIACC, Universidade do Porto, Portugal

    Ana Paula Tomás & Miguel Filgueiras

Authors
  1. Ana Paula Tomás
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  2. Miguel Filgueiras
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Hubert Comon

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© 1997 Springer-Verlag Berlin Heidelberg

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Tomás, A.P., Filgueiras, M. (1997). Solving linear Diophantine equations using the geometric structure of the solution space. In: Comon, H. (eds) Rewriting Techniques and Applications. RTA 1997. Lecture Notes in Computer Science, vol 1232. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-62950-5_77

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  • DOI: https://doi.org/10.1007/3-540-62950-5_77

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-62950-4

  • Online ISBN: 978-3-540-69051-1

  • eBook Packages: Springer Book Archive

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