An Application of Gro ̈bner Basis

Vimala Ramani

doi:10.32628/IJSRST24114309

Authors

Vimala Ramani Department of Mathematics, Anna University, Chennai, Tamil Nadu, India Author

DOI:

https://doi.org/10.32628/IJSRST24114309

Keywords:

Multivariate polynomial, monomial order, Gro ̈bner basis, height of an element, ideal membership problem

Abstract

We prove that the height of x_1 is 16 in A=Q[x_1,x_2,y_1,…,y_8 ]/I, where deg(x_i)=deg(y_i )=2i, and I is the ideal generated by the relation (1+x_1 +x_2 )(1 +y_1+⋯+y_8 )=1, using a Gro ̈bner basis technique. We discuss the topological implication and significance of computing the height of x_1.

Downloads

Download data is not yet available.

References

T. Becker and V.Weispfenning, “Gro ̈bner Bases: A Computational Approach to Commutative Algebra,” Graduate Texts in Mathematics, Springer-Verlag, New York (1993). 3. DOI: https://doi.org/10.1007/978-1-4612-0913-3

Borel A.(1953). Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts. Ann. of Math. (2). 1953; 57 (2):115-207. https://people.math.rochester.edu/faculty/doug/otherpapers/Borel-Sur.pdf DOI: https://doi.org/10.2307/1969728

Colović U.A., Prvulović B. I., Cup-length of oriented Grassmann manifolds via Gr¨obner bases, J. Algebra 642 (2024) 256–285. https://doi.org/10.1016/j.jalgebra.2023.11.040 DOI: https://doi.org/10.1016/j.jalgebra.2023.11.040

Colović U. A., Prvulović B. I., Radovanović M. Topologicial complexity of oriented Grassmann manifolds. arXiv:2402.13336. 2024: https://doi.org/10.48550/arXiv.2402.13336

Fukaya T. Gr¨obner bases of oriented Grassmann manifolds, Homol. Homotopy Appl. 10:2 (2008) 195–209. http://dx.doi.org/10.4310/HHA.2008.v10.n2.a10 DOI: https://doi.org/10.4310/HHA.2008.v10.n2.a10

Korbas ̌ J, Lo ̈rinc J. The Z_2-cohomology cup-length of real flag manifolds. Fund. Math. 2003; 178(2) :143-158. https://eudml.org/doc/283181 DOI: https://doi.org/10.4064/fm178-2-4

Petrović Z Z, Prvulović B I, Radovanović M. Multiplication in the cohomology of Grassmannians via Gröbner bases. J. Algebra. 2015; 438:60-84. https://doi.org/10.1016/j.jalgebra.2015.04.031 DOI: https://doi.org/10.1016/j.jalgebra.2015.04.031

Petrović Z.Z., Prvulović B.I., Radovanović M., Gröbner bases for (partial) flag manifolds, Journal of Symbolic Computation, (2020); (101): 90-108, https://doi.org/10.1016/j.jsc.2019.06.008. DOI: https://doi.org/10.1016/j.jsc.2019.06.008

Prvulović B. I. Gröbner bases for complex Grassmann manifolds. Publ. Inst. Math. 2011;90(104):23–46. https://doi.org/10.2298/PIM1104023P DOI: https://doi.org/10.2298/PIM1104023P

Radovanović M. On the topological complexity and zero-divisor cup-length of real Grassmannians. Proceedings of the Royal Society of Edinburgh. 2023;153:702–717. https://doi.org/10.1017/prm.2022.15 DOI: https://doi.org/10.1017/prm.2022.15

Ramani V, Sankaran P. On degrees of maps between Grassmannians. Proc. Indian Acad. Sci. (Math. Sci.) 1997;107(1):13-19. https://doi.org/10.1007/BF02840469 DOI: https://doi.org/10.1007/BF02840469

Ramani V. On the topological complexity of Grassmann manifolds. Math. Slovaca. 2020;70(5): 1197-1210. https://doi.org/10.1515/ms-2017-0425 DOI: https://doi.org/10.1515/ms-2017-0425

Ramani V. The rational zero-divisor cup-length of oriented partial flag manifolds. Math. Slovaca. 2023; 73 (5):1325-1357. https://doi.org/10.1515/ms-2023-0097 DOI: https://doi.org/10.1515/ms-2023-0097

M. Stillman, “Grobner Bases: a Tutorial,” Lecture Notes. https://www.risc.jku.at/Groebner-Bases-Bibliography/gbbib_files/publication_190.pdf