## Abstract

Standard bases of ideals of the polynomial ring *R*[*X*] = *R*[*x*
_{1}, …, *x*
_{
k
}] over a commutative Artinian chain ring *R* that are concordant with the norm on *R* have been investigated by D. A. Mikhailov, A. A. Nechaev, and the author. In this paper we continue this investigation. We introduce a new order on terms and a new reduction algorithm, using the coordinate decomposition of elements from *R*. We prove that any ideal has a unique reduced (in terms of this algorithm) standard basis. We solve some classical computational problems: the construction of a set of coset representatives, the finding of a set of generators of the syzygy module, the evaluation of ideal quotients and intersections, and the elimination problem. We construct an algorithm testing the cyclicity of an LRS-family *L*
_{
R
}(*I*), which is a generalization of known results to the multivariate case. We present new conditions determining whether a Ferre diagram \(\mathcal{F}\) and a full system of \(\mathcal{F}\)-monic polynomials form a shift register. On the basis of these results, we construct an algorithm for lifting a reduced Gröbner basis of a monic ideal to a standard basis with the same cardinality.

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## References

- 1.
W. Adams and P. Loustaunau,

*An Introduction to Gröbner Bases*, American Mathematical Society, Providence (1994), Graduate Studies in Mathematics, Vol. 3. - 2.
J. Apel, “Computational ideal theory infinitely generated extension rings,”

*J. Theoretical Computer Science*,**244**, 1–33 (2000). - 3.
M. F. Atiyah and I. G. MacDonald,

*Introduction to Commutative Algebra*, Addison-Wesley (1969). - 4.
E. Byrne and P. Fitzpatrick, “Gröbner bases over Galois rings with an application to decoding alternant codes,”

*J. Symbolic Comput.*,**31**, 565–584 (2001). - 5.
D. Cox, J. Little, and D. O’Shea,

*Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra*, Springer, Berlin-New York (1992). - 6.
C. Faith,

*Algebra II. Ring Theory*, Springer, Berlin (1976). - 7.
E. V. Gorbatov, “Standard basis of a polynomial ideal over commutative Artinian chain ring,”

*Discrete Math. Appl.*,**14**, No. 1, 75–101 (2004). - 8.
E. V. Gorbatov and A. A. Nechaev, “A criterion of the cyclicity of a family of linear recurring sequences over a QF-module,”

*Usp. Mat. Nauk*,**56**, No. 4 (2001). - 9.
F. Kasch,

*Moduln und Ringe*, Teubner, Stuttgart (1977). - 10.
L. Kronecker,

*Vorlesungen über Zahlentheorie*, Bd. 1, Teubner, Leipzig (1901). - 11.
W. Krull, “Algebraische Theorie der Ringe, II,”

*Math. Ann.*,**91**, 1–46 (1923). - 12.
V. L. Kurakin, A. S. Kuzmin, A. V. Mikhalev, and A. A. Nechaev, “Linear recurring sequences over rings and modules,”

*J. Math. Sci.*,**76**, No. 6, 2793–2915 (1995). - 13.
V. L. Kurakin, A. V. Mikhalev, A. A. Nechaev, and V. N. Tsypyschev, “Linear and polylinear recurring sequences over Abelian groups and modules,”

*J. Math. Sci.*,**102**, No. 6, 4598–4626 (2000). - 14.
V. N. Latyshev,

*Combinatorial Ring Theory, Standard Bases*[in Russian], Izd. Mosk. Univ., Moscow (1988). - 15.
P. Lu,

*A Criterion for Annihilating Ideals of Linear Recurring Sequences over Galois Rings*, AAECC-417. - 16.
D. A. Mikhailov and A. A. Nechaev, “Canonical generating system of a monic polynomial ideal over a commutative Artinian chain ring,”

*Diskret. Mat.*,**13**, No. 4, 3–42 (2001). - 17.
D. A. Mikhailov and A. A. Nechaev, “Solving systems of polynomial equations over Galois-Eisenstein rings with the use of the canonical generating systems of polynomial ideals,”

*Discrete Math. Appl.*,**14**, No. 1, 41–73 (2004). - 18.
T. Mora,

*Seven Variations on Standard Bases*, Preprint, Univ. de Genova, Dip. di Mathematica, No. 45 (1986). - 19.
A. A. Nechaev, “Linear recurring sequences over commutative rings,”

*Diskret. Mat.*,**3**, No. 4, 105–127 (1991). - 20.
A. A. Nechaev, “Linear recurring sequences over quasi-Frobenius modules,”

*Russian Math. Surveys*,**48**, No. 3 (1993). - 21.
A. A. Nechaev, “Finite quasi-Frobenius modules, applications to codes and linear recurrences,”

*Fund. Prikl. Mat.*,**1**, No. 1, 229–254 (1995). - 22.
A. A. Nechaev, “Polylinear recurring sequences over modules and quasi-Frobenius modules,” in:

*Proc. First Int. Tainan-Moscow Algebra Workshop, 1994*, Walter de Gruyter, Berlin-New York (1996), pp. 283–298. - 23.
G. H. Norton and A. Salagean, “Strong Gröbner bases and cyclic codes over a finite-chain ring,” in:

*Proceedings of the International Workshop on Coding and Cryptography*, Paris (2001), pp. 8–12. - 24.
L. Robbiano, “On the theory of graded structures,”

*J. Symbolic Comput.*,**2**, 139–170 (1986).

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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 10, No. 3, pp. 23–71, 2004.

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Gorbatov, E.V. Standard bases concordant with the norm and computations in ideals and polylinear recurring sequences.
*J Math Sci* **139, **6672–6707 (2006). https://doi.org/10.1007/s10958-006-0384-3

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### Keywords

- Standard Basis
- Shift Register
- Polynomial Ideal
- Polynomial System
- Coset Representative