解位于少数子空间的乘法群中的未知数线性方程
Linear equations with unknowns from a multiplicative group whose solutions lie in a small number of subspaces
摘要：
设K是特征为0的域。我们考虑线性方程a1*x1+.+ an*xn=1，未知数x1，...，xn，其中a1，...，an是K的非零元素，并且其中G是K的非零元素的乘法群的子群。两个元组（a1，...，an）和（b1，...，bn）称为G-等价的，如果存在u1，...，un在G中，使得b1=a1*u1，...，bn=an*un。由m（a1，...，an，G）最小数m使得a1*x1+.+的解的集合an*xn=1 in x1，.，xn包含在K^n的m个真线性子空间的并中。已知m（a1，...，an，G）是有限的;显然，如果（a1，...，an）被替换为G等价元组。Gyory和作者在1988年证明了存在一个常数c（n），它只取决于变量n的个数，使得对于除了n之外的所有G-等价类（a1，...，an），一个具有m（a1，...，an，G）< c（n）。目前还不清楚c（n）的最佳可能值是多少。Gyory和作者证明了c（n）=2^{（n+1）！}可以被带走。作者在1993年将其改进为c（n）=（n！）^ {2n+2}。本文将其进一步改进为c（n）=2^{n+1}，并举例说明c（n）不可能小于n.
英文摘要：
Let K be a field of characteristic 0. We consider linear equations a1*x1+...+an*xn=1 in unknowns x1,...,xn from G, where a1,...,an are non-zero elements of K, and where G is a subgroup of the multiplicative group of non-zero elements of K. Two tuples (a1,...,an) and (b1,...,bn) of non-zero elements of K are called G-equivalent if there are u1,...,un in G such that b1=a1*u1,..., bn=an*un. Denote by m(a1,...,an,G) the smallest number m such that the set of solutions of a1*x1+...+an*xn=1 in x1,...,xn from G is contained in the union of m proper linear subspaces of K^n. It is known that m(a1,...,an,G) is finite; clearly, this quantity does not change if (a1,...,an) is replaced by a G-equivalent tuple. Gyory and the author proved in 1988 that there is a constant c(n) depending only on the number of variables n, such that for all but finitely many G-equivalence classes (a1,...,an), one has m(a1,...,an,G)< c(n). It is as yet not clear what is the best possible value of c(n). Gyory and the author showed that c(n)=2^{(n+1)!} can be taken. This was improved by the author in 1993 to c(n)=(n!)^{2n+2}. In the present paper we improve this further to c(n)=2^{n+1}, and give an example showing that c(n) can not be smaller than n.
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