Additive Relations in Fields: An Entropy Approach Original Research Article

Let ξ1, ...,ξr be complex numbers with K = image(ξ1, ...,ξr) having transcendence degree r − 1 over image. Consider the equation a1x1 + · · · + akxk = 1, in which the ai′s are fixed elements of K×, no proper subsum ailxil + · · · + aijxij vanishes, and we seek solutions xi set membership, variant Γ = left angle bracket, doubleξ1, ..., ξrright angle bracket, double. It is well-known that (1) has only finitely many solutions; we present here an elementary proof of this fact using results from the entropy theory of commuting group automorphisms.