Hostname: page-component-848d4c4894-8bljj Total loading time: 0 Render date: 2024-06-23T18:35:41.867Z Has data issue: false hasContentIssue false
  • English
  • Français

On the set of Hilbert polynomials

Published online by Cambridge University Press:  17 April 2009

Alexander B. Levin
Alexander B. Levin
Affiliation:
Department of Mathematics, The Catholic University of America, Washington DC 20064, United States of America
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We characterise the set of all Hilbert polynomials of standard graded algebras over a field and give solutions of some open problems on Hilbert polynomials. In particular, we prove that a chromatic polynomial of a graph is a Hilbert polynomial of some standard graded algebra.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 64 , Issue 2 , October 2001 , pp. 291 - 305
Copyright
Copyright © Australian Mathematical Society 2001

References

[1]Brenti, F., ‘Hilbert polynomials in combinatorics’, J. Algebraic Combin. 7 (1998), 127–156.CrossRefGoogle Scholar
[2]Cox, D., Little, J. and O'Shea, D., Ideals, varieties, and algorithms. An introduction to computational algebraic geometry and commutative algebra (Springer-Verlag, Berlin, Heidelberg, New York, 1992).Google Scholar
[3]Hilbert, D., ‘Über die theorie der algebraischen formen’, Math. Annal. 36 (1890), 473–534.CrossRefGoogle Scholar
[4]Johnson, J.L., ‘Differential dimension polynomials and a fundamental theorem on differential modules’, Amer. J. Math. 91 (1969), 239–248.CrossRefGoogle Scholar
[5]Kolchin, E.R., ‘The notion of dimension in the theory of algebraic differential equations’, Bull Amer. Math. Soc. 70 (1964), 570–573.CrossRefGoogle Scholar
[6]Kolchin, E.R., Differential algebra and algebraic groups (Academic Press, New York, London, 1973).Google Scholar
[7]Kondrateva, M.V., Levin, A.B., Mikhalev, A.V. and Pankratev, E.V., Differential and difference dimension polynomials (Kluwer Academic Publishers, Dordrecht, 1999).CrossRefGoogle Scholar
[8]Levin, A.B., ‘Characteristic polynomials of filtered difference modules and of difference field extensions’, Russian Math. Surveys 33 (1978), 165–166.CrossRefGoogle Scholar
[9]Levin, A.B., ‘Characteristic polynomials of inversive difference modules and some properties on inversive difference dimension’, Russian Math. Surveys 35 (1980), 217–218.CrossRefGoogle Scholar
[10]Macaulay, F.S., ‘Some properties of enumeration in the theory of modular systems’, Proc. London Math. Soc.(2) 26 (1927), 531–555.CrossRefGoogle Scholar
[11]Read, R.C., ‘An introduction to chromatic polynomials’, J. Combin. Theory 4 (1968), 52–71.CrossRefGoogle Scholar
[12]Zariski, O. and Samuel, P., Commutative Algebra, Vol. II (D. Van Nostard Co., Inc., London, 1960).CrossRefGoogle Scholar