What the study found
The paper gives examples involving elliptic and quasi-elliptic functions for which algebraic independence is proved. These examples include z, the Weierstrass function wp(z), zeta function zeta(z), sigma function sigma(z), exponential functions, and Serre functions related to integrals of the third kind.
Why the authors say this matters
The authors state that these examples fit a strong version of Schanuel's Conjecture, a statement about algebraic independence among complex numbers and their function values. The study suggests this is relevant because similar statements can hold when the exponential function is replaced by algebraically independent functions.
What the researchers tested
The article studies tuples of complex numbers and compares them with values of several transcendental functions. It focuses on algebraic independence statements, especially Schanuel-type assertions, for elliptic and quasi-elliptic functions.
What worked and what didn't
The abstract says that for almost all tuples (x1, …, xn) of complex numbers, the 2n numbers x1, …, xn, e^x1, …, e^xn are algebraically independent. It also says that similar statements hold for other algebraically independent functions, and that the paper gives examples where the listed elliptic, quasi-elliptic, exponential, and Serre functions are proved to be algebraically independent.
What to keep in mind
The abstract does not describe the proof details, specific conditions beyond “almost all” tuples, or any limitations of the examples beyond the functions named.
Key points
- The paper presents examples where elliptic and quasi-elliptic functions are algebraically independent.
- Named functions include z, wp(z), zeta(z), sigma(z), exponential functions, and Serre functions related to integrals of the third kind.
- The abstract says that for almost all complex tuples (x1, …, xn), the numbers x1, …, xn, e^x1, …, e^xn are algebraically independent.
- The authors connect these examples to a strong version of Schanuel's Conjecture.
- The abstract does not give proof details or specific limitations beyond the named scope.
Disclosure
- Research title:
- Elliptic and quasi-elliptic functions satisfy Schanuel-type independence
- Authors:
- Michel Waldschmidt
- Institutions:
- Sorbonne Université
- Publication date:
- 2026-04-27
- OpenAlex record:
- View
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