Tropical Cramer Determinants Revisited

We prove general Cramer type theorems for linear systems over various extensions of the tropical semiring, in which tropical numbers are en- riched with an information of multiplicity, sign, or argument. We obtain exis- tence or uniqueness results, which extend or rene earlier results of Gondran and Minoux (1978), Plus (1990), Gaubert (1992), Richter-Gebert, Sturmfels and Theobald (2005) and Izhakian and Rowen (2009). Computational issues are also discussed; in particular, some of our proofs lead to Jacobi and Gauss- Seidel type algorithms to solve linear systems in suitably extended tropical semirings.

[1]  ENUMERATIVE TROPICAL ALGEBRAIC GEOMETRY IN R , 2004 .

[2]  Zur Izhakian,et al.  Supertropical algebra , 2008, 0806.1171.

[3]  Richard A. Brualdi,et al.  Matrices of Sign-Solvable Linear Systems , 1995 .

[4]  S. Gaubert,et al.  Linear independence over tropical semirings and beyond , 2008, 0812.3496.

[5]  G. Mikhalkin.  Enumerative tropical algebraic geometry , 2003 .

[7]  B. Carré.  An Algebra for Network Routing Problems , 1971 .

[8]  Oleg Viro,et al.  Dequantization of Real Algebraic Geometry on Logarithmic Paper , 2000, math/0005163.

[9]  Mauro Dell'Amico,et al.  Assignment Problems , 1998, IFIP Congress: Fundamentals - Foundations of Computer Science.

[10]  Douglas Lind,et al.  Non-archimedean amoebas and tropical varieties , 2004, math/0408311.

[11]  Z. Izhakian,et al.  The Tropical Rank of a Tropical Matrix , 2006, math/0604208.

[12]  P..  Regularity of matrices in min-algebra and its time-complexity , 2003 .

[13]  V. Kolokoltsov,et al.  Idempotent Analysis and Its Applications , 1997 .

[14]  S. Gaubert,et al.  Asymptotics of the Perron eigenvalue and eigenvector using Max-algebra , 1998 .

[15]  R. Dedekind.  Über die Theorie der ganzen algebraischen Zahlen , 1964 .

[16]  Zur Izhakian.  Tropical Arithmetic and Matrix Algebra , 2009 .

[17]  Peter Butkovic,et al.  Strong Regularity of Matrices - A Survey of Results , 1994, Discret. Appl. Math..

[18]  Manfred Knebusch,et al.  Categories of Layered Semirings , 2012, 1207.3488.

[19]  Виктор Павлович Маслов,et al.  Идемпотентный функциональный анализ. Алгебраический подход@@@Idempotent Functional Analysis: An Algebraic Approach , 2001 .

[20]  Eugenii Shustin,et al.  Tropical Algebraic Geometry , 2007 .

[21]  U. Zimmermann.  Linear and combinatorial optimization in ordered algebraic structures , 1981 .

[22]  B. Sturmfels,et al.  First steps in tropical geometry , 2003, math/0306366.

[23]  M. Plus.  Linear systems in (inax,+) algebra , 1990 .

[24]  M. Gondran,et al.  Algèbre linéaire et cheminement dans un graphe , 1975 .

[25]  M. Yoeli.  A Note on a Generalization of Boolean Matrix Theory , 1961 .

[26]  Michel Minoux,et al.  Graphes et algorithmes , 1995 .

[27]  Oleg Viro.  Hyperfields for Tropical Geometry I. Hyperfields and dequantization , 2010 .

[28]  Bernd Sturmfels,et al.  Maximal Minors and Their Leading Terms , 1993 .

[29]  Alexander E. Guterman,et al.  Tropical Polyhedra are Equivalent to mean Payoff Games , 2009, Int. J. Algebra Comput..

[30]  William J. Cook,et al.  Combinatorial optimization , 1997 .

[31]  M. Joswig.  Tropical Halfspaces , 2003, math/0312068.

[32]  Maja Ilc.  On the rank of a tropical matrix , 2012 .