# Algebraic independence of elementary functions and its application to Masser's vanishing theorem

Research Papers

- Received:
- Accepted:

DOI: 10.1007/BF02112281

- Cite this article as:
- Nishioka, K. & Nishioka, K. Aeq. Math. (1990) 40: 67. doi:10.1007/BF02112281

## Summary

Here is an improvement on Masser's Refined Identity (D. W. Masser:. If

*A vanishing theorem for power series.*Invent. Math.*67*(1982), 275–296). The present method depends on a result from differential algebra and*p*-adic analysis. The investigation from the viewpoint of*p*-adic analysis makes the proof clearer and, in particular, it is possible to exclude the concept of “density” which is necessary in Masser's treatment. That is to say, the theorem will be stated as follows: Let Ω = (*ω*_{ij}) be a nonsingular matrix in*M*_{n}(ℤ) with no roots of unity as eigenvalue. Let*P(z)*be a nonzero polynomial inC[*z*],*z*= (*z*_{1},⋯,*z*_{n}). Let*x*= (*x*_{1},⋯,*x*_{n}) be an element of*C*^{n}with*x*_{i}≠ 0 for each*i*. Define$$\Omega x = \left( {\prod\limits_{i = 1}^n {x_i^{\omega 1_i } ,...,} \prod\limits_{i = 1}^n {x_i^{\omega _{ni} } } } \right)$$

*P*(Ω^{k}*x*) = 0 for infinitely many positive integers*k*, then*x*_{1},⋯,*x*_{n}are multiplicatively dependent.To prove this, the following fact on elementary functions will be needed: Let*K* be an ordinary differential field and*C* be its field of constants. Let*R* be a differential field extension of*K* and*u*_{1},⋯,*u*_{m} be elements of*R* such that the field of constants of*R* is the same as*C* and for each*i* the field extension*K*_{i} =*K*(*u*_{1},⋯,*u*_{i}) of*K* is a differential one such that*u*′_{i} =*t*′_{i−1}*u*_{i} for some*t*_{i−1}∈*K*_{i−1} or*u*_{i} is algebraic over*K*_{i−1}. Let*f*_{1},⋯,*f*_{n} ∈*R* be distinct elements modulo*C* and suppose that for each*i* there is a nonzero*e*_{i} ∈*R* with*e′*_{i} =*f′*_{i}*e*_{i}. Then*e*_{1},⋯,*e*_{n} are linearly independent over*K.*

### AMS (1980) subject classification

Primary 11J81Secondary 12H05## Copyright information

© Birkhäuser Verlag 1990