Border bases are an alternative to Gröbner bases. The former have several more desirable properties. In this paper some constructions and operations on border bases are presented. Namely; the case of a restriction of an ideal to a polynomial ring (in a smaller number of variables), the case of the intersection of two ideals, and the case of the kernel of a homomorphism of polynomial rings. These constructions are applied to the ideal of relations and to factorizable derivations.
@article{bwmeta1.element.doi-10_2478_s11533-010-0045-0,
author = {Janusz Zieli\'nski},
title = {Border bases and kernels of homomorphisms and of derivations},
journal = {Open Mathematics},
volume = {8},
year = {2010},
pages = {780-785},
zbl = {1200.13041},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-010-0045-0}
}
Janusz Zieliński. Border bases and kernels of homomorphisms and of derivations. Open Mathematics, Tome 8 (2010) pp. 780-785. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-010-0045-0/
[1] Chen Y.F., Meng X.H., Border bases of positive dimensional polynomial ideals, In: Proceedings of the 2007 International Workshop on Symbolic-Numeric Computation, London, Ontario, July 25–27, ACM, New York, 2007, 65–71
[2] Gianni P., Trager B., Zacharias G., Gröbner bases and primary decomposition of polynomial ideals, J. Symbolic Comput., 1988, 6(2–3), 149–167 http://dx.doi.org/10.1016/S0747-7171(88)80040-3[Crossref] | Zbl 0667.13008
[3] Kehrein A., Kreuzer M., Characterizations of border bases, J. Pure Appl. Algebra, 2005, 196(2–3), 251–270 http://dx.doi.org/10.1016/j.jpaa.2004.08.028[Crossref] | Zbl 1081.13011
[4] Kehrein A., Kreuzer M., Computing border bases, J. Pure Appl. Algebra, 2006, 205(2), 279–295 http://dx.doi.org/10.1016/j.jpaa.2005.07.006[Crossref]
[5] Kehrein A., Kreuzer M., Robbiano L., An algebraist’s view on border bases, In: Solving polynomial equations, Algorithms Comput. Math., 14, Springer, Berlin, 2005, 169–202 http://dx.doi.org/10.1007/3-540-27357-3_4[Crossref] | Zbl 1152.13304
[6] Kreuzer M., Robbiano L., Computational Commutative Algebra, 1&2, Springer, Berlin, 2000&2005 http://dx.doi.org/10.1007/978-3-540-70628-1[Crossref]
[7] Nowicki A., Zielinski J., Rational constants of monomial derivations, J. Algebra, 2006, 302(1), 387–418 http://dx.doi.org/10.1016/j.jalgebra.2006.02.034[Crossref] | Zbl 1119.13021
[8] Zieliński J., Factorizable derivations and ideals of relations, Comm. Algebra, 2007, 35(3), 983–997 http://dx.doi.org/10.1080/00927870601117639[WoS][Crossref] | Zbl 1171.13013