Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-09T19:33:23.119Z Has data issue: false hasContentIssue false

An Improved Result Concerning Singular Manifolds of Difference Polynomials

Published online by Cambridge University Press:  20 November 2018

Richard M. Cohn*
Affiliation:
Rutgers University
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.

Let be a difference field of characteristic 0, m an irreducible manifold of effective order n over {y}, and F an algebraically irreducible difference polynomial in {y} of effective order n + k, k > 0, which vanishes on 3 m. In an earlier paper (2, p. 447) I gave necessary conditions, restated below as (a), (b), and (c) of the main theorem, for m to be an essential singular manifold of F. These conditions are analogous to the low power criterion of Ritt (1, p. 65) for the corresponding problem of differential algebra. Like that criterion they depend, in the special case that m is the manifold of y, only on which power products appear effectively in F. Unlike the low power criterion, however, conditions (a), (b), and (c) are only necessary, not sufficient.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1959

References

1. Ritt, J. F., Differential algebra, Coll. publ. Amer. Math. Soc, 33. Google Scholar
2. Cohn, R. M., Singular manifolds of difference polynomials, Ann. Math., 53 (1951), 445–63.Google Scholar
3. Cohn, R. M., Extensions of difference fields, Amer. J. Math., 74 (1952), 507-30.Google Scholar
4. Cohn, R. M., Essential singular manifolds of difference polynomials, Ann. Math., 57 (1953), 524-30.Google Scholar
5. Hodge, W. V. D. and Pedoe, D., Methods of algebraic geometry (Cambridge University Press).Google Scholar