schanuel猜想

Introduction

The Schanuel conjecture, also known as the Lindemann–Weierstrass conjecture, is a proposition in transcendental number theory. It asserts that if α1, α2, …, αn are linearly independent complex numbers, then the transcendence degree over the field of rational numbers of the set of numbers {α1,e^α1,α2,e^α2, …, αn,e^αn} is at least n. This conjecture was proposed by Jean-Pierre Serre in 1972, and it remains an open problem in mathematics.

Background

The Schanuel conjecture is closely related to the Lindemann–Weierstrass theorem, which states that if α1, α2, …, αn are algebraic numbers which are linearly independent over the field of rational numbers, then the numbers {e^α1, e^α2, …, e^αn} are also linearly independent over the field of algebraic numbers. This theorem has important applications in various fields, such as the proof of the transcendence of π and certain values of the Riemann zeta function.

Statement of the conjecture

The Schanuel conjecture extends the Lindemann–Weierstrass theorem by considering not only algebraic numbers, but also transcendental numbers. Let K be a field of characteristic 0, and let α1, α2, …, αn be linearly independent over K. Denote by L the field generated by K and the numbers {α1,e^α1,α2,e^α2, …, αn,e^αn}. Then the transcendence degree over K of L is at least n.

Current status of the conjecture

Despite decades of efforts, the Schanuel conjecture remains unsolved. However, several special cases and related results have been established. For example, the conjecture is known to be true when n ≤ 2. Moreover, it has been proved that the transcendence degree of the set {π,e^π,π^2,e^π^2} over the field of rational numbers is at least 4, which is a remarkable consequence of the Lindemann–Weierstrass theorem.

Various methods have been used to attack the Schanuel conjecture, including algebraic geometry, number theory, and model theory. The most promising approach seems to be the theory of exponential diophantine equations, which studies the solutions of equations involving exponential functions. For example, the conjecture is equivalent to the statement that the exponential function is diophantine in the field generated by a given set of transcendental numbers over the field of rational numbers.

Significance of the conjecture

The Schanuel conjecture has important implications in various areas of mathematics and science. For example, it is related to the length of continued fractions and the distribution of prime numbers. It also has applications in computer science, physics, and engineering, such as the design of error-correcting codes and the study of quantum field theory.

Moreover, the Schanuel conjecture is a fundamental problem in the theory of transcendental numbers, which has a rich history and deep connections with other branches of mathematics. It is one of the few remaining major conjectures in this field, and its solution would shed light on many aspects of transcendence and algebraic independence.

Conclusion

The Schanuel conjecture is a challenging and important problem in transcendental number theory, which asserts that certain sets of transcendental numbers are algebraically independent over rational fields. Despite decades of research, the conjecture remains unsolved, but many related results and approaches have been developed. Its solution would have profound implications in various fields of mathematics and science, and it is an outstanding problem in the theory of transcendental numbers.