Small solutions to systems of polynomial equations with integer coe-cients
Mihai Cipu · 2011
The paper discusses a series of conjectures due to A. Tyszka aiming to describe boxes in which there exists at least one solution to a system of polynomial equations with integer coe‐cients. A proof of the bound valid in the linear case is given. 1 Two basic questions When facing systems of equations whose solutions are hard to determine, one is satisfled to determine (or at least estimate) the number and the size of solutions. A satisfactory answer could be an algorithm, if a deflnite formula is unavailable. These questions are completely answered only for univariate polynomials over the ring of integers or the fleld of rational, real or complex numbers. Many important results, such as Falting’s result on rational points on irreducible algebraic curves of genus at least 2, ensures the flniteness of the solution set to speciflc systems without giving any hint on its cardinality. A great deal of mathematics appeared as a result of attempts to solve such ‡ i : ¶ ;