On a class of Diophantine equations related to the numbers of cells in hyperplane arrangements
The author studies the diophantine equation \[ {x\choose m}+\dots+{x\choose 1}={y\choose n}+\dots+{y\choose 1} \] in integers \(x\geq m\) and \(y\geq n\), where \(m\) and \(n\) are distinct integers greater than \(1\). The equation can be nicely interpreted geometrically: one asks when the numbers of cells in two simple hyperplane arrangements are equal. The author proves that for any fixed \(n>m\geq 3\) the above equation has only finitely many solutions. The main method of the proof is a deep theorem of Bilu and Tichy. Further, it is also proved that if \(m\geq 3\) and \(n=2\) then the soultions \(x,y\) can be effectively bounded in terms of \(m\). Here the main tool is a previous result of the author concerning the root structures of certain polynomials and the Baker-method. As it is pointed out, in fact the methods of the proofs are capable to handle much more general equations, as well.
scientific article; zbMATH DE number 5565744