Introduction to Algebraic Independence TheoryYuri V. Nesterenko, Patrice Philippon In the last five years there has been very significant progress in the development of transcendence theory. A new approach to the arithmetic properties of values of modular forms and theta-functions was found. The solution of the Mahler-Manin problem on values of modular function j(tau) and algebraic independence of numbers pi and e^(pi) are most impressive results of this breakthrough. The book presents these and other results on algebraic independence of numbers and further, a detailed exposition of methods created in last the 25 years, during which commutative algebra and algebraic geometry exerted strong catalytic influence on the development of the subject. |
Contents
| 1 | |
Mahlers conjecture and other transcendence results | 13 |
Algebraic independence for values of Ramanujan functions | 27 |
Some remarks on proofs of algebraic independence | 47 |
Elimination multihomogene | 52 |
Diophantine geometry | 83 |
Geometrie diophantienne multiprojective | 95 |
Criteria for algebraic independence | 132 |
Zero Estimates on Commutative Algebraic Groups | 166 |
Measures of algebraic independence for Mahler functions | 187 |
Small Transcendence Degrees | 198 |
Large Transcendence Degrees | 213 |
Some metric results in Transcendental Numbers Theory | 226 |
The Hilbert Nullstellensatz Inequalities for Polynomials and Algebraic Independence | 239 |
| 249 | |
| 254 | |
Upper bounds for geometric Hilbert functions | 143 |
Multiplicity estimates for solutions of algebraic differential equations | 149 |
Other editions - View all
Introduction to Algebraic Independence Theory Yuri V. Nesterenko,Patrice Philippon No preview available - 2014 |
Introduction to Algebraic Independence Theory, Issue 1752 Patrice Philippon No preview available - 2001 |
Common terms and phrases
algebraic group algebraic independence algebraic number algebraic subgroup Ann(M assertion assume bien chapitre Chapter Chow form Chu2 coefficients common zeros commutative algebraic complex numbers conjecture constant Corollary d'indice definition degré DÉMONSTRATION denote differential equations dimension Dist(x donne e j(M Eisenstein series élément elliptic functions exists exponential exponential map fait finie forme résultante formule hauteur Hilbert function homogeneous ideal homogeneous polynomial idéal implies independent over Q inequality infinie integer irreducible l'idéal l'on Lemma lemme linear linearly independent log(n lower bound modular form modular function monôme montre morphisme multidegré multigradué multihomogène nonzero notations Notons Nullstellensatz open neighborhood p-adic peut Pn(C polynôme PPh7 premier prime ideal produit projective variety Proposition proved real number result satisfies subset sufficiently large suivant Suppose Theorem 1.1 théorème tout transcendence degree unmixed ideal upper bound vanishes variables voir zéro zero estimate