Homogeneous Sagbi Bases Under Polynomial Composition

Nazish Kanwal


The process of replacing indeterminates in a Polynomial with other polynomials is the polynomial composition. Homogeneous Sagbi bases are the Sagbi bases generated by the subset of homogeneous polynomials. In this article we present adequate and essential criterion on a set of polynomials to guarantee that the composed set \(S\circ \ominus\) is Homogeneous Sagbi basis whenever \(S\) is a Homogeneous Sagbi basis.


Homogeneoous Sagbi basis; Polynomial composition

Full Text:



B. Buchberger, An Algorithm for Finding the Bases of the Residue Class Ring Modulo a Zero Dimensional Polynomial Ideal, Ph.D. Thesis, Universitat Innsbruck, Austria (1965), URL: https://core.ac.uk/download/pdf/82459944.pdf.

B. Buchberger, Gröbner bases: an algorithmic method in polynomial ideal theory, in Multidimensional Systems Theory – Progress, Directions and Open Problems in Multidimensional Systems, N. K. Bose (editor), Reidel Publishing Company, 184 – 232 (1985), DOI: 10.1007/978-94-009-5225-6_6.

H. Hong, Gröebner, Basis under composition II, in Proceeding ISSAC ’96 Proceedings of the 1996 International Symposium on Symbolic and Algebraic Computation Zurich, Switzerland, July 24-26, 1996, pp. 79 – 85, ACM Press (1996), DOI: 10.1145/236869.236906.

H. Hong, Gröebner basis under composition I, J. Symbolic Computation 25 (1998), 643 – 662, DOI: 10.1006/jsco.1997.0192.

J. A. Khan, Further on the composition of Sagbi bases, International Electronic Journal of Algebra 20 (2016), 100 – 110, URL: http://www.ieja.net/files/papers/volume-20/6-V20-2016.pdf.

J. Liu and M. Wang, Homogeneous Gröebner bases under composition, Journal of Algebra 303 (2006), 668 – 676, DOI: 10.1016/j.jalgebra.2005.08.037.

J. Liu and M. Wang, Further results on homogeneous Gröebner bases under composition, Journal of Algebra 315 (2007), 134 – 143, DOI: 10.1016/j.jalgebra.2007.05.023.

L. Robbiano, On the theory of graded structures, J. Symbolic Computation 2(2) (1986), 139 – 170, DOI: 10.1016/S0747-7171(86)80019-0.

L. Robbiano and M. Sweedler, Subalgebra basses, in Commutative Algebra (Proceedings of a Workshop held in Salvador, Brazil, August 8-17, 1988), W. Bruns and A. Simis (eds.), Vol. 1430 of Lecture Notes in Mathematics Series, Springer-Verlag, 42, 61 – 87 (1988), https://www.springer.com/us/book/9783540527459.

L. Robbiano and M. Sweedler, Subalgebra bases in commutative algebra, in Lecture notes in Mathematics 1430, W. Bruns and A. Simis (eds.), 61 – 87, Springer, Berlin — Heidelberg (1990), DOI: 10.1007/BFb0085537.

P. Nordbeck, Sagbi bases under composition, J. Symbolic Computation 33 (2002), 67 – 76, DOI: 10.1006/jsco.2001.0498.

DOI: http://dx.doi.org/10.26713%2Fcma.v10i3.1219


  • There are currently no refbacks.

eISSN 0975-8607; pISSN 0976-5905