Book contents
- Frontmatter
- Contents
- Acknowledgements
- 1 Introduction
- 2 Diophantine classes: definitions and basic facts
- 3 Diophantine equivalence and Diophantine decidability
- 4 Integrality at finitely many primes and divisibility of order at infinitely many primes
- 5 Bound equations for number fields and their consequences
- 6 Units of rings of W-integers of norm 1
- 7 Diophantine classes over number fields
- 8 Diophantine undecidability of function fields
- 9 Bounds for function fields
- 10 Diophantine classes over function fields
- 11 Mazur's conjectures and their consequences
- 12 Results of Poonen
- 13 Beyond global fields
- Appendix A Recursion (computability) theory
- Appendix B Number theory
- References
- Index
9 - Bounds for function fields
Published online by Cambridge University Press: 14 October 2009
- Frontmatter
- Contents
- Acknowledgements
- 1 Introduction
- 2 Diophantine classes: definitions and basic facts
- 3 Diophantine equivalence and Diophantine decidability
- 4 Integrality at finitely many primes and divisibility of order at infinitely many primes
- 5 Bound equations for number fields and their consequences
- 6 Units of rings of W-integers of norm 1
- 7 Diophantine classes over number fields
- 8 Diophantine undecidability of function fields
- 9 Bounds for function fields
- 10 Diophantine classes over function fields
- 11 Mazur's conjectures and their consequences
- 12 Results of Poonen
- 13 Beyond global fields
- Appendix A Recursion (computability) theory
- Appendix B Number theory
- References
- Index
Summary
In this chapter we will discuss some bound equations specialized for function fields. These bounds will be used in the next chapter in our discussion of Diophantine classes of function fields. Some methods used below should be familiar to the reader from Chapter 5.
Height bounds
In this section we will consider how to obtain information about the height of a function, given information on the height of a polynomial evaluated at this function. We also compare the height of the coordinates of a field element with respect to a chosen basis and the height of the element itself. (The reader is reminded that the definition of the height of a function field element can be found in B.1.25.)
Lemma 9.1.1. Let K be a function field and let F(T) ∈ K[T] be a polynomial of degree greater than or equal to 1. Let HK(x) denote the height of x in K. Then there exists a positive constant CF, depending on F(T) only, such that for all x ∈ K we have that HK(x) ≤ CF·(HK(F(x)).
Proof. Since the case where the degree of F(T) is equal to 1 is obvious, we will assume that the degree of F(T) is greater than 1.
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- Information
- Hilbert's Tenth ProblemDiophantine Classes and Extensions to Global Fields, pp. 162 - 165Publisher: Cambridge University PressPrint publication year: 2006