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

Keiji Nishioka, Kumiko Nishioka

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2 Citations (Scopus)


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 = (ωij) be a nonsingular matrix in Mn (ℤ) with no roots of unity as eigenvalue. Let P(z) be a nonzero polynomial in C[z], z = (z1,⋯, zn). Let x = (x1,⋯, xn) be an element of Cn with xi ≠ 0 for each i. Define {Mathematical expression}. If P(Ωkx) = 0 for infinitely many positive integers k, then x1,⋯, xn 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 u1,⋯, um be elements of R such that the field of constants of R is the same as C and for each i the field extension Ki =K(u1,⋯, ui) of K is a differential one such that u′i =t′i-1ui for some ti-1∈Ki-1 or ui is algebraic over Ki-1. Let f1,⋯, fn ∈R be distinct elements modulo C and suppose that for each i there is a nonzero ei ∈R with e′i =f′iei. Then e1,⋯, en are linearly independent over K.

Original languageEnglish
Pages (from-to)67-77
Number of pages11
JournalAequationes Mathematicae
Issue number1
Publication statusPublished - 1990 Dec 1


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

ASJC Scopus subject areas

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


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