Sagbi bases over the product of rings

Year 2024, Early Access, 1 - 11

Nazia Jabeen , Junaid Khan

https://doi.org/10.24330/ieja.1575706

Abstract

Let $R$ be a commutative ring such that $R=R_1\times \cdots \times R_n$. In this paper, we give a method to compute (strong) Sagbi bases for subalgebras of a polynomial ring over $R$ provided that these bases are computable in a polynomial ring over $R_i$ for $1\leq i \leq n$. As an application, we prove the existence of strong Sagbi bases for subalgebras in a polynomial ring over a principal ideal ring.

Keywords

Gröbner basis, strong Gröbner basis, Sagbi basis, PIR, PID

References

• W. W. Adams, S. Hosten, P. Loustaunau and J. L. Miller, Sagbi and Sagbi-Gröbner bases over principal ideal domains, J. Symbolic Comput., 27(1) (1999), 31-47.
• W. W. Adams and P. Loustaunau, An Introduction to Gröbner Bases, Graduate Studies in Mathematics, 3, American Mathematical Society, Providence, RI, 1994.
• B. Buchberger, An Algorithm for Finding the Bases of the Residue Class Ring Modulo a Zero Dimensional Polynomial Ideal, Ph.D. thesis, University of Innsbruck, Austria, 1965.
• B. Buchberger and F. Winkler, Gröbner Bases and Applications, London Mathematical Society Lecture Note Series, 251, Cambridge University Press, Cambridge, 1998.
• M. Kreuzer and L. Robbiano, Computational Commutative Algebra 1, Springer-Verlag, Berlin, 2000.
• J. L. Miller, Analogs of Gröbner bases in polynomial rings over a ring, J. Symbolic Comput., 21(2) (1996), 139-153.
• G. H. Norton and A. Salagean, Strong Gröbner bases for polynomials over a principal ideal ring, Bull. Austral. Math. Soc., 64(3) (2001), 505-528.
• G. H. Norton and A. Salagean, Gröbner bases and products of coefficient rings, Bull. Austral. Math. Soc., 65(1) (2002), 145-152.
• L. Robbiano and M. Sweedler, Subalgebra bases, Commutative algebra (Salvador, 1988), Lecture Notes in Math., Springer, Berlin, 1430 (1990), 61-87.
• M. Stillman and H. Tsai, Using Sagbi bases to compute invariants, J. Pure Appl. Algebra, 139(1-3) (1999), 285-302.
• O. Zariski and P. Samuel, Commutative Algebra, Vol. 1, Graduate Texts in Mathematics, 28, Springer-Verlag, New York-Heidelberg-Berlin, 1975.