### Abstract

Here is an improvement on Masser's Refined Identity (D. W. Masser: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 in C[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 {Mathematical expression}. If 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.

Original language | English |
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Pages (from-to) | 67-77 |

Number of pages | 11 |

Journal | Aequationes Mathematicae |

Volume | 40 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1990 Dec 1 |

### Keywords

- AMS (1980) subject classification: Primary 11J81, Secondary 12H05

### ASJC Scopus subject areas

- Mathematics(all)
- Discrete Mathematics and Combinatorics
- Applied Mathematics

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## Cite this

*Aequationes Mathematicae*,

*40*(1), 67-77. https://doi.org/10.1007/BF02112281