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- Title
Extensions of Büchi's Higher Powers Problem to Positive Characteristic.
- Authors
Pasten, Hector; Julie Tzu-Yueh Wang
- Abstract
Büchi's nth power problem on Q asks whether there exist an integer M such that the only monic polynomials F∈Q[X] of degree n satisfying that F(1),...,F(M) are nth power rational numbers, are precisely of the form F(X)=(X+c)n for some c∈Q. In this paper, we study analogs of this problem for algebraic function fields of positive characteristic. We formulate and prove an analog (indeed, such a formulation for n>2 was missing in the literature due to some unexpected phenomena), which we use to derive some definability and undecidability consequences. Moreover, in the case of characteristic zero, we extend some known results by improving the bounds for M (from quadratic on n to linear on n).
- Subjects
POLYNOMIALS; INTEGERS; RATIONAL numbers; ALGEBRAIC functions; MATHEMATICAL bounds
- Publication
IMRN: International Mathematics Research Notices, 2015, Vol 2015, Issue 11, p3263
- ISSN
1073-7928
- Publication type
Article
- DOI
10.1093/imrn/rnu033