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Linear equations with unknowns from a multiplicative group whose solutions lie in a small number of subspaces
Linear equations with unknowns from a multiplicative group whose solutions lie in a small number of subspaces
- Source :
- Indagationes Mathematicae. 15(3):347-355
- Publication Year :
- 2004
- Publisher :
- Elsevier BV, 2004.
-
Abstract
- 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.<br />12 pages, latex file
- Subjects :
- Discrete mathematics
11D61
Mathematics(all)
Mathematics - Number Theory
Multiplicative group
Exponential equations
General Mathematics
Field (mathematics)
Cartesian product
Linear subspace
Combinatorics
Multiplication (music)
symbols.namesake
symbols
FOS: Mathematics
Rank (graph theory)
Number Theory (math.NT)
Tuple
Linear equations wich unknowns from a multiplicativegroup
Linear equation
Mathematics
Subjects
Details
- ISSN :
- 00193577
- Volume :
- 15
- Issue :
- 3
- Database :
- OpenAIRE
- Journal :
- Indagationes Mathematicae
- Accession number :
- edsair.doi.dedup.....53371fa2b574365d14e398fbcd5a8b32
- Full Text :
- https://doi.org/10.1016/s0019-3577(04)80004-1