Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The growth of valuations on rational function fields in two variables

Authors: Edward Mosteig and Moss Sweedler
Journal: Proc. Amer. Math. Soc. 132 (2004), 3473-3483
MSC (2000): Primary 13F30, 13F25; Secondary 13P10
Published electronically: July 20, 2004
MathSciNet review: 2084067
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Given a valuation on the function field $k(x,y)$, we examine the set of images of nonzero elements of the underlying polynomial ring $k[x,y]$ under this valuation. For an arbitrary field $k$, a Noetherian power series is a map $z:\mathbb{Q}\to k$ that has Noetherian (i.e., reverse well-ordered) support. Each Noetherian power series induces a natural valuation on $k(x,y)$. Although the value groups corresponding to such valuations are well-understood, the restrictions of the valuations to underlying polynomial rings have yet to be characterized. Let $\Lambda_n$ denote the images under the valuation $v$ of all nonzero polynomials $f \in k[x,y]$of at most degree $n$ in the variable $y$. We construct a bound for the growth of $\Lambda_n$ with respect to $n$ for arbitrary valuations, and then specialize to valuations that arise from Noetherian power series. We provide a sufficient condition for this bound to be tight.

References [Enhancements On Off] (What's this?)

  • 1. S. S. Abhyankar and T. T. Moh, Newton-Puiseux expansion and generalized Tschirnhausen transformation, part 1, J. Reine Angew. Math 260 (1973) 47-83. MR 49:2724
  • 2. Dominique Duval, Rational Puiseux Series, Compositio Mathematica 70 (1989) 119-154. MR 90c:14001
  • 3. Olav Geil and Ruud Pellikaan, On the Structure of Order Domains, Finite Fields and Their Applications 9 (2002) 369-396. MR 2003i:13034
  • 4. H. Hahn, Über die nichtarchimedischen Größensysteme, Sitz. Akad. Wiss. Wien 116 (1907) 601-655.
  • 5. Kiran Kedlaya, The Algebraic Closure of the Power Series Field in Positive Characteristic, Proceedings of the American Mathematical Society 129 (2001) 3461-3470. MR 2003a:13025
  • 6. Saunders MacLane and O.F.G. Schilling, Zero-Dimensional Branches of Rank One on Algebraic Varieties, Annals of Mathematics 40 (1939) 507-520. MR 1:26c
  • 7. Edward Mosteig, A Valuation-Theoretic Approach to Polynomial Computations, Doctoral Thesis, Cornell University, 2000.
  • 8. Edward Mosteig, Value Monoids of Zero-Dimensional Valuations of Rank One, in preparation.
  • 9. Edward Mosteig and Moss Sweedler, Valuations and Filtrations, Journal of Symbolic Computation 34 (2002), no. 5, 399-435. MR 2003j:12008
  • 10. Edward Mosteig, Computing Leading Exponents of Noetherian Power Series, Communications in Algebra 30 (2002) 6055-6069. MR 2003j:13030
  • 11. Michael E. O'Sullivan, New Codes for the Berlekamp-Massey-Sakata Algorithm, Finite Fields and Their Applications 7 (2001) 293-317. MR 2002b:94050
  • 12. Moss Sweedler, Ideal Bases and Valuation Rings, manuscript, 1986, available at http://math.
  • 13. Oscar Zariski, The reduction of the singularities of an algebraic surface, Annals of Mathematics 40 (1939) 639-689. MR 1:26d

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 13F30, 13F25, 13P10

Retrieve articles in all journals with MSC (2000): 13F30, 13F25, 13P10

Additional Information

Edward Mosteig
Affiliation: Department of Mathematics, Loyola Marymount University, Los Angeles, California 90045

Moss Sweedler
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853

Keywords: Valuations, generalized power series, Gr\"obner bases
Received by editor(s): January 10, 2002
Received by editor(s) in revised form: July 14, 2003
Published electronically: July 20, 2004
Communicated by: Michael Stillman
Article copyright: © Copyright 2004 American Mathematical Society