Diophantine Analysis: Proceedings at the Number Theory Section of the 1985 Australian Mathematical Society Convention (London Mathematical Society Lecture Note Series, Series Number 109) [1 ed.] 0521339235, 9780521339230

The papers in this volume, which were presented at the 1985 Australian Mathematical Society convention, survey recent wo

127 105 980KB

English Pages 184 [181] Year 1986

Table of contents :
Cover
Series Page
Title
Copyright
CONTRIBUTORS
CONTENTS
INTRODUCTION
AN IDEAL TRANSCENDENCE MEASURE
A. INTRODUCTION
B. DEFINITIONS AND STATEMENT OF RESULTS
C. REDUCTION OF THEOREM 1 TO THE PRIME CASE
D. THUE-SIEGEL-NESTERENKO LEMMA
E. THE AUXILIARY FUNCTION
F. THE ZERO ESTIMATE
G. UPPER BOUND FOR IR(e.,e..)
H. LOWER BOUND FOR IR(e.,e..)
I. DEDUCTION OF COROLLARIES
J. REMARK
REFERENCES
GALOIS ORBITS ON ABELIAN VARIETIES AND ZERO ESTIMATES
1. INTRODUCTION
2. SCHNEIDER'S METHOD ([7])
3. BAKER'S METHOD
REFERENCES
SMALL SOLUTIONS OF CONGRUENCES WITH PRIME MODULUS
1. INTRODUCTION
2. QUOTING A LEMMA ON EXPONENTIAL SUMS
3. PROOF OF THEOREM 1
4. AN APPLICATION OF THE GEOMETRY OF NUMBERS
5. THE CASE d = 2 OF THEOREM 2
6. THE CASE d = 3 OF THEOREM 3
7. CLASSIFICATION OF SYSTEMS OF FORMS IN TERMS OF COMPOSITION
8. PROOF OF PROPOSITION 1
9. A WEAK VERSION OF THEOREM 3
10. PROOF OF (1.8) AND OF THEOREM 3
APPENDIX
REFERENCES
NEWTON POLYHEDRA AND SOLUTIONS OF CONGRUENCES
1. INTRODUCTION
2. THE NEWTON POLYHEDRON
3. THE INDICATOR DIAGRAM
4. COMMON ZEROS
REFERENCES
ON PRIME FACTORS OF SUMS OF INTEGERS II
1. NOTATION
2. THE CASE IAI = IBI = 2
3. LOWER BOUNDS FOR w(II1) AND w(II2): ELEMENTARY PROOFS
4. LOWER BOUNDS FOR w(II1) AND w(II2): NON-ELEMENTARY PROOFS
5. UPPER BOUNDS FOR w(II1) AND w(II2)
6. LOWER BOUNDS FOR P(II1) AND P(II2)
7. UPPER BOUNDS FOR P(II1) AND P(II2)
REFERENCES
AN INTRODUCTION TO CONTINUED FRACTIONS
1. Notice that if
2. We turn now to various formal properties of continued fractions which are immediate consequences of the correspondence:
3. PERIODIC CONTINUED FRACTIONS
4. MULTIPLYING A CONTINUED FRACTION (BY A RATIONAL)
REFERENCES
SEARCH FOR THE THREE DIMENSIONAL APPROXIMATION CONSTANT
REFERENCES
INVERSE PROBLEMS FOR MAHLER'S MEASURE
1. INTRODUCTION
2. RESTRICTIONS ON THE SIZE OF MEASURES
3. THE PISOT-VIJAYARAGHAVAN AND SALEM NUMBERS
4. MEASURES OF NONRECIPROCALS
5. RESTRICTIONS ON MEASURES
6. FURTHER RESTRICTIONS ON MEASURES
7. RECIPROCAL NUMBERS WITH NON-RECIPROCAL MEASURES
8. CONCLUDING REMARKS
REFERENCES
LARGE NEWMAN POLYNOMIALS
1. INTRODUCTION
2. THE GROWTH OF .(n)
3. COMPUTATION OF .(n) FOR SMALL n
4. ESTIMATES OF .(n) FOR LARGE n
5. ESTIMATES OF .
6. CONCLUSIONS AND CONJECTURES
REFERENCES
APPENDIX A

Diophantine Analysis: Proceedings at the Number Theory Section of the 1985 Australian Mathematical Society Convention (London Mathematical Society Lecture Note Series, Series Number 109) [1 ed.]
0521339235, 9780521339230

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