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Gröbner Bases in Difference-Differential Modules And Science in China Series A: Mathematics Sep., 2008, Vol. 51, No. 9, 1732{1752 www.scichina.com math.scichina.com GrÄobnerbases in di®erence-di®erential modules and di®erence-di®erential dimension polynomials ZHOU Meng1y & Franz WINKLER2 1 Department of Mathematics and LMIB, Beihang University, and KLMM, Beijing 100083, China 2 RISC-Linz, J. Kepler University Linz, A-4040 Linz, Austria (email: [email protected], [email protected]) Abstract In this paper we extend the theory of GrÄobnerbases to di®erence-di®erential modules and present a new algorithmic approach for computing the Hilbert function of a ¯nitely generated di®erence- di®erential module equipped with the natural ¯ltration. We present and verify algorithms for construct- ing these GrÄobnerbases counterparts. To this aim we introduce the concept of \generalized term order" on Nm £ Zn and on di®erence-di®erential modules. Using GrÄobnerbases on di®erence-di®erential mod- ules we present a direct and algorithmic approach to computing the di®erence-di®erential dimension polynomials of a di®erence-di®erential module and of a system of linear partial di®erence-di®erential equations. Keywords: GrÄobnerbasis, generalized term order, di®erence-di®erential module, di®erence- di®erential dimension polynomial MSC(2000): 12-04, 47C05 1 Introduction The e±ciency of the classical GrÄobnerbasis method for the solution of problems by algorithmic way in polynomial ideal theory is well-known. The results of Buchberger[1] on GrÄobnerbases in polynomial rings have been generalized by many researchers to non-commutative case, especially to modules over various rings of di®erential operators. Galligo[2] ¯rst gave the GrÄobnerbasis [3] algorithm for the Weyl algebra An. Mora generalized the concept of GrÄobnerbasis to non- [4] [5] commutative free algebras. Noumi and Takayama formulated the GrÄobnerbases in Rn, the ring of di®erential operators with rational function coe±cients. Oaku and Shimoyama[6] treated [7] D0, the ring of di®erential operators with power series coe±cients. Insa and Pauer presented a basic theory of GrÄobnerbases for di®erential operators with coe±cients in a commutative noetherian ring. It has been proved that the notion of GrÄobnerbasis is a powerful tool to solve various problems of linear partial di®erential equations. On the other hand, for some problems of linear di®erence-di®erential equations such as the dimension of the space of solutions and the computation of di®erence-di®erential dimension polynomials, the notion of GrÄobnerbasis for the ring of di®erence-di®erential operators is essential. GrÄobnerbases in rings of di®erential operators are de¯ned with respect to a term order Received May 24, 2007; accepted February 22, 2008 DOI: 10.1007/s11425-008-0081-4 y Corresponding author This work was supported by the National Natural Science Foundation of China (Grant No. 60473019) and the KLMM (Grant No. 0705) GrÄobnerbases in di®erence-di®erential modules and di®erence-di®erential dimension polynomials 1733 on Nn £Nn or Nn. This approach cannot be used for the ring of di®erence-di®erential operators, because for it we need to treat orders on Nm£Zn. Pauer and Unterkircher[8] considered GrÄobner bases in Laurent polynomial rings, but it is limited in commutative case. Levin[9] introduced a characteristic set for free modules over rings of di®erence-di®erential operators. It is an analog of \GrÄobnerbasis" connected with a speci¯c \ordering" on Nm £ Zn. But the ordering is not a term-ordering while the theory of GrÄobnerbasis works for any term-ordering. The concept of the di®erential dimension polynomial was introduced by Kolchin[10] as a dimensional description of some di®erential ¯eld extension. Johnson[11] proved that the di®er- ential dimension polynomial of a di®erential ¯eld extension coincides with the Hilbert polyno- mial of some ¯ltered di®erential module. This result allowed to compute di®erential dimension polynomials using the GrÄobnerbasis technique. Since then various problems of di®erential al- gebra involving di®erential dimension polynomials were studied (see [12, 13]). The concepts of the di®erence dimension polynomial and the di®erence-di®erential dimension polynomial were introduced ¯rst in [14, 15] respectively. They play the same role in di®erence algebra (resp. di®erence-di®erential algebra) as Hilbert polynomials in commutative algebra or di®erential dimension polynomials in di®erential algebra. The notion of di®erence-di®erential dimension polynomial can be used for the study of dimension theory of di®erence-di®erential ¯eld extension and of systems of algebraic di®erence-di®erential equations. Mikhalev and Pankratev[15] proved existence of di®erence-di®erential dimension polynomials Á(t) associated with a di®erence-di®erential module M by classical GrÄobnerbasis methods of computation of Hilbert polynomials. The proof is based on the fact that the ring of di®erence- di®erential operators over the di®erence-di®erential ¯eld R is isomorphic to the factor ring of the ring of generalized polynomials R[x1; : : : ; xm+2n] (where xia = axi+±i(a) (1 6 i 6 m), xm+ja = ¡1 ®j(a)xm+j and xm+n+ja = ®j (a)xm+n+j (1 6 j 6 n) for any a 2 R) by the ideal I generated [16] by the polynomials xm+jxm+n+j ¡ 1 (1 6 j 6 n). Wu also computed dimensions of linear di®erence-di®erential systems by the above approach. However, a similar approach to di®erence- di®erential dimension polynomials in two variables is unsuccessful. Levin[9] investigated the di®erence-di®erential dimension polynomials in two variables by the characteristic set approach. The method of Levin is rather delicate but no general algorithm for computing the characteristic set. In this paper we present a new approach, di®erence-di®erential GrÄobnerbasis, for algorith- mic computing the di®erence-di®erential dimension polynomials. It is based on the algorithm of computation of GrÄobnerbasis for an ideal of (or a module over) the ring of di®erence- di®erential operators. In Section 2 the generalized term order and its properties are discussed and some examples are presented. In Section 3 we design carefully the reduction algorithm, the de¯nition of the GrÄobnerbasis and the S-polynomials, as well as the Buchberger algorithm for the computation of the GrÄobnerbases. In Section 4 we describe an approach to com- puting di®erence-di®erential dimension polynomials associated with a module over the ring of di®erence-di®erential operators via the GrÄobnerbases. Throughout the paper Z, N, Z¡ and Q denotes the sets of all integers, all non-negative integers, all non-positive integers, and all rational numbers, respectively. By a ring we always mean an associative ring with a unit. By the module over a ring A we mean a unitary left 1734 ZHOU Meng & Franz WINKLER A-module. De¯nition 1.1. Let R be a commutative noetherian ring, ¢ = f±1; : : : ; ±mg and σ = f®1; : : : ; ®ng be set of derivations and automorphisms of the ring R, respectively, such that ¯(x) 2 R and ¯(γ(x)) = γ(¯(x)) hold for any ¯; γ 2 ¢ [ σ and x 2 R. Then R is called a di®erence-di®erential ring with the basic set of derivations ¢ and the basic set of automorphisms σ, or shortly a ¢-σ-ring. If R is a ¯eld, then it is called a ¢-σ-¯eld. If R is a ¢-σ-ring, then ¤ will denote the commutative semigroup of elements of the form k1 km l1 ln ¸ = ±1 ¢ ¢ ¢ ±m ®1 ¢ ¢ ¢ ®n ; (1:1) m n where (k1; : : : ; km) 2 N and (l1; : : : ; ln) 2 Z . This semigroup contains the free commutative semigroup £ generated by the set ¢ and free commutative semigroup ¡ generated by the set ¡1 ¡1 ¤ σ. The subset f®1; : : : ; ®n; ®1 ; : : : ; ®n g of ¤ will be denoted by σ . De¯nition 1.2. Let R be a ¢-σ-ring and the semigroup ¤ be as above. Then an expression of the form X a¸¸; (1:2) ¸2¤ where a¸ 2 R for all ¸ 2 ¤ and only ¯nitely many coe±cients a¸ are di®erent from zero, is called a di®erence-di®erential operator (or shortly a ¢-σ-operator) over R. Two ¢-σ-operators P P ¸2¤ a¸¸ and ¸2¤ b¸¸ are equal if and only if a¸ = b¸ for all ¸ 2 ¤. The set of all ¢-σ-operators over a ¢-σ-ring R is a ring with the following fundamental relations X X X µ X ¶ X a¸¸ + b¸¸ = (a¸ + b¸)¸; a a¸¸ = (aa¸)¸; ¸2¤ ¸2¤ ¸2¤ ¸2¤ ¸2¤ µ X ¶ X (1.3) a¸¸ ¹ = a¸(¸¹); ±a = a± + ±(a); ¿a = ¿(a)¿; ¸2¤ ¸2¤ ¤ ¤ for all a¸; b¸ 2 R, ¸; ¹ 2 ¤, a 2 R, ± 2 ¢, ¿ 2 σ . Note that the elements in ¢ and σ do not commute with the elements in R, and then the \terms" ¸ 2 ¤ do not commute with the coe±cients a¸ 2 R. De¯nition 1.3. The ring of all ¢-σ-operators over a ¢-σ-ring R is called the ring of di®erence-di®erential operators (or shortly the ring of ¢-σ-operators) over R, which will be denoted by D. A left D-module M is called a di®erence-di®erential module (or a ¢-σ-module): If M is ¯nitely generated as a left D-module, then M is called a ¯nitely generated ¢-σ-module. When σ = ;, D will be the rings of di®erential operators R[±1; : : : ; ±m]. If the coe±cient ring R is the polynomial ring over a ¯eld K, then D will be Weyl algebra Am. So ¢-σ-module is a generalization of module over rings of di®erential operators.
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