Standard bases in mixed power series and polynomial rings over rings

In this paper we study standard bases for submodules of a mixed power series and polynomial ring R ź t 1 , ź , t m ź x 1 , ź , x n s respectively of their localisation with respect to a t _ -local monomial ordering for a certain class of noetherian rings R, also called Zacharias rings. The main steps are to prove the existence of a division with remainder generalising and combining the division theorems of Grauert-Hironaka and Mora and to generalise the Buchberger criterion. Everything else then translates naturally. Setting either m = 0 or n = 0 we get standard bases for polynomial rings respectively for power series rings over R as a special case.

[1]  Eimear Byrne,et al.  Gröbner Bases over Commutative Rings and Applications to Coding Theory , 2009, Gröbner Bases, Coding, and Cryptography.

[2]  Hans Schönemann,et al.  Standard bases, syzygies and their implementation in SINGULAR , 1994 .

[3]  Gerhard Pfister,et al.  Advances and improvements in the theory of standard bases and syzygies , 1996 .

[4]  Rekha R. Thomas,et al.  Computing tropical varieties , 2007, J. Symb. Comput..

[5]  Deepak Kapur,et al.  Computing a Gröbner Basis of a Polynomial Ideal over a Euclidean Domain , 1988, J. Symb. Comput..

[6]  Ferdinando Mora,et al.  An Algorithm to Compute the Equations of Tangent Cones , 1982, EUROCAM.

[7]  Hans Grauert,et al.  Über die deformation isolierter singularitäten analytischer mengen , 1971 .

[8]  Afshan Sadiq,et al.  Standard Bases over Rings , 2009, Int. J. Algebra Comput..

[9]  Deepak Kapur,et al.  Algorithms for Computing Groebner Bases of Polynomial Ideals over Various Euclidean Rings , 1984, EUROSAM.

[10]  Michael Kalkbrener,et al.  Converting Bases with the Gröbner Walk , 1997, J. Symb. Comput..

[11]  G. Greuel,et al.  A Singular Introduction to Commutative Algebra , 2002 .

[12]  Rekha R. Thomas,et al.  Computing Gröbner fans , 2007, Math. Comput..

[13]  Thomas Markwig,et al.  Standard bases in K , 2008, J. Symb. Comput..

[14]  D. Bayer.  The division algorithm and the hilbert scheme , 1982 .

[15]  Thomas Markwig,et al.  A Field of Generalised Puiseux Series for Tropical Geometry , 2007, 0709.3784.

[16]  Teo Mora,et al.  La queste del saint Gra(AL): A computational approach to local algebra , 1991, Discret. Appl. Math..

[17]  Yue Ren,et al.  Gröbner fans of x-homogeneous ideals in R〚t〛[x] , 2017, J. Symb. Comput..

[18]  Teo Mora,et al.  The Gröbner Fan of an Ideal , 1988, J. Symb. Comput..

[19]  Hans-Gert Gräbe,et al.  The tangent cone algorithm and homogenization , 1994 .

[20]  Alessandro Logar.  Constructions over localizations of rings , 1989 .

[21]  H. Hauser,et al.  EFFECTIVE ALGEBRAIC POWER SERIES , 2005 .

[22]  H. Michael Möller,et al.  On the Construction of Gröbner Bases Using Syzygies , 1988, J. Symb. Comput..

[23]  Komei Fukuda,et al.  The generic Gröbner walk , 2007, J. Symb. Comput..

[24]  Teo Mora,et al.  A computational model for algebraic power series , 1992 .

[25]  I. Yengui,et al.  Dynamical Gröbner bases over Dedekind rings , 2010 .

[26]  Daniel Lichtblau,et al.  Effective computation of strong Gröbner bases over Euclidean domains , 2012 .

[27]  W. T. Tutte,et al.  Encyclopedia of Mathematics and its Applications , 2001 .

[28]  W. W. Adams,et al.  An Introduction to Gröbner Bases , 2012 .

[29]  Herwig Hauser,et al.  Perfect bases for differential equations , 2005, J. Symb. Comput..

[30]  T. Becker.  Stability and buchberger criterion for standard bases in power series rings , 1990 .