On the diophantine equation \(\sum ^{n}_{i=1}x_ i/d_ i\equiv 0\) (mod 1) and its applications
Let \(A(d_ 1,d_ 2,...,d_ n)\) denote the number of solutions in integers \(x_ 1,x_ 2,...,x_ n\) of the equation \[ \sum^{n}_{i=1}x_ i/d_ i\equiv 0\quad (mod 1)\quad with\quad 0<x_ i<d_ i\quad (i=1,2,...,n), \] where \(d_ 1,d_ 2,...,d_ n\) are given positive integers. The quantity \(A(d_ 1,d_ 2,...,d_ n)\) is of significance in the study of the number of solutions \(x_ 1,x_ 2,...,x_ n\) in the finite field GF(q) of the equation \(\sum^{n}_{i=1}a_ i x_ i^{d_ i}=0\) with coefficients in GF(q). It has been shown by \textit{W. M. Schmidt} [cf. Lemma 6D in Chapter IV of his book, Equations over finite fields. An elementary approach (Lect. Notes Math. 536) (1976; Zbl 0329.12001)] that \[ A(d,d,...,d)=\frac{d-1}{d}[(d-1)^{n-1}-(-1)^{n-1}]. \] In this paper the authors prove by the inclusion-exclusion principle that one has \[ A(d_ 1,d_ 2,...,d_ n)=\sum^{n}_{k=1}(-1)^{n-k}\sum_{1\leq i_ 1<i_ 2<...<i_ k\leq n}\frac{d_{i_ 1}d_{i_ 2}...d_{i_ k}}{[d_{i_ 1},d_{i_ 2},...,d_{i_ k}]}+(-1)^ n, \] where [a,b,...,c] denotes the least common multiple of the positive integers a,b,...,c. Some special cases of this formula are also discussed.
scientific article; zbMATH DE number 3955000
scientific article