Finiteness theorems and algorithms for permutation invariant chains of Laurent lattice ideals

We study chains of lattice ideals that are invariant under a symmetric group action. In our setting, the ambient rings for these ideals are polynomial rings which are increasing in (Krull) dimension. Thus, these chains will fail to stabilize in the traditional commutative algebra sense. However, we prove a theorem which says that ''up to the action of the group'', these chains locally stabilize. We also give an algorithm, which we have implemented in software, for explicitly constructing these stabilization generators for a family of Laurent toric ideals involved in applications to algebraic statistics. We close with several open problems and conjectures arising from our theoretical and computational investigations.

[1]  Gregory Minton,et al.  Voting, the Symmetric Group, and Representation Theory , 2007, Am. Math. Mon..

[2]  Anton Leykin,et al.  Noetherianity for infinite-dimensional toric varieties , 2013, 1306.0828.

[3]  D. Eisenbud.  Commutative Algebra: with a View Toward Algebraic Geometry , 1995 .

[4]  Akimichi Takemura,et al.  A Markov basis for two-state toric homogeneous Markov chain model without initial parameters , 2010, 1005.1717.

[5]  B. Sturmfels,et al.  Binomial Ideals , 1994, alg-geom/9401001.

[6]  Andrew Snowden,et al.  Syzygies of Segre embeddings and $\Delta$-modules , 2010, 1006.5248.

[7]  Joseph B. Kruskal,et al.  The Theory of Well-Quasi-Ordering: A Frequently Discovered Concept , 1972, J. Comb. Theory A.

[8]  Claudiu Raicu.  Secant varieties of Segre–Veronese varieties , 2010, 1011.5867.

[9]  Daniel E. Cohen,et al.  Closure Relations, Buchberger's Algorithm, and Polynomials in Infinitely Many Variables , 1987, Computation Theory and Logic.

[10]  C. Nash-Williams.  On well-quasi-ordering infinite trees , 1963, Mathematical Proceedings of the Cambridge Philosophical Society.

[11]  Andrew Snowden,et al.  Syzygies of Segre embeddings , 2010 .

[12]  Alberto Vigneron-Tenorio,et al.  On decomposable semigroups and applications , 2013, J. Symb. Comput..

[13]  Seth Sullivant,et al.  Finite Groebner bases in infinite dimensional polynomial rings and applications , 2009, 0908.1777.

[14]  Graham Higman,et al.  Ordering by Divisibility in Abstract Algebras , 1952 .

[15]  B. Sturmfels,et al.  Combinatorial Commutative Algebra , 2004 .

[16]  Bernd Sturmfels,et al.  Higher Lawrence configurations , 2003, J. Comb. Theory, Ser. A.

[17]  Ruriko Yoshida,et al.  Degree Bounds for a Minimal Markov Basis for the Three-State Toric Homogeneous Markov Chain Model , 2011, 1108.0481.

[18]  Viktor Levandovskyy,et al.  Letterplace ideals and non-commutative Gröbner bases , 2009, J. Symb. Comput..

[19]  Alan R. Camina,et al.  SOME INFINITE PERMUTATION MODULES , 1991 .

[20]  Matthias Aschenbrenner,et al.  ERRATUM FOR FINITE GENERATION OF SYMMETRIC IDEALS , 2009 .

[21]  Tsit Yuen Lam,et al.  A first course in noncommutative rings , 2002 .

[22]  Chee-Keng Yap,et al.  Fundamental problems of algorithmic algebra , 1999 .

[23]  Eric H. Kuo.  Viterbi sequences and polytopes , 2006, J. Symb. Comput..

[24]  Anton Leykin,et al.  Equivariant lattice generators and Markov bases , 2014, ISSAC.

[25]  Matthias Aschenbrenner,et al.  Finite generation of symmetric ideals , 2004, math/0411514.

[26]  Seth Sullivant,et al.  Toric ideals of phylogenetic invariants. , 2005, Journal of computational biology : a journal of computational molecular cell biology.

[27]  Ivar Ugi,et al.  Die Vandermondesche Determinante als Näherungsansatz für eine Chiralitätsbeobachtung, ihre Verwendung in der Stereochemie und zur Berechnung der optischen Aktivität , 1967 .

[28]  B. Sturmfels.  Gröbner bases and convex polytopes , 1995 .

[29]  Michiel Hazewinkel,et al.  Algebras, rings and modules , 2004 .

[30]  Seth Sullivant,et al.  A finiteness theorem for Markov bases of hierarchical models , 2007, J. Comb. Theory, Ser. A.

[31]  J. M. Landsberg,et al.  On the Ideals of Secant Varieties of Segre Varieties , 2004, Found. Comput. Math..

[32]  Martin Ziegler,et al.  Quasi finitely axiomatizable totally categorical theories , 1986, Ann. Pure Appl. Log..

[33]  Andries E. Brouwer,et al.  Equivariant Gröbner bases and the Gaussian two-factor model , 2011, Math. Comput..

[34]  David A. Cox,et al.  Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics) , 2007 .

[35]  Claudiu Raicu.  The GSS Conjecture , 2010 .

[36]  Jan Draisma,et al.  Bounded-rank tensors are defined in bounded degree , 2011, 1103.5336.

[37]  M. Kazarian,et al.  KP hierarchy for Hodge integrals , 2008, 0809.3263.

[38]  D. E Cohen,et al.  On the laws of a metabelian variety , 1967 .

[39]  M. Drton,et al.  Algebraic factor analysis: tetrads, pentads and beyond , 2005, math/0509390.

[40]  Jan Draisma,et al.  Finiteness for the k-factor model and chirality varieties , 2008, 0811.3503.

[41]  David Mumford,et al.  What Can Be Computed in Algebraic Geometry , 1993, alg-geom/9304003.

[42]  Jan Draisma,et al.  On the ideals of equivariant tree models , 2007, 0712.3230.

[43]  Akimichi Takemura,et al.  Markov chain Monte Carlo test of toric homogeneous Markov chains , 2010, 1004.3599.

[44]  Donal O'Shea,et al.  Ideals, varieties, and algorithms - an introduction to computational algebraic geometry and commutative algebra (2. ed.) , 1997, Undergraduate texts in mathematics.

[45]  Satoshi Aoki,et al.  Minimal and minimal invariant Markov bases of decomposable models for contingency tables , 2010 .