Fractional parts of linear polynomials and an application to hypergeometric functions (Q2747268)

Let \(\{x\}\) denote the fractional part of \(x\) and put \(((x))=\{x\}-\frac 12\). Let \(L_j\) be linear polynomials in \(\mathbb Q(\mathbf y)\), where \(\mathbf y=(y_1,\dots,y_h)\), and let \(c_0,c_1,\dots\) be rational numbers. The authors consider relations of the type \(c_0+\sum_jc_j((L_j(\mathbf y)))=0\). In particular, they investigate rational \(h\)-tuples \(\pmb\rho= (\rho_1,\dots,\rho_h)\) with the following property: Let \(m\) be the common denominator of the \(\rho_i\). Then \(\mathbf y=n\pmb\rho\) is required to be a solution of this relation for every \(n\) prime to \(m\). They show that such vectors \(\pmb\rho\) form a ``quasi-linear'' set which is described by finitely many linear equalities modulo \(\mathbb Z^h\) and can be effectively computed. NEWLINENEWLINENEWLINEThere is an interesting connection with an old theorem of Schwarz giving the conditions for the classical hypergeometric differential equation to have algebraic solutions. These conditions can be translated in the above form, and the authors compute explicitly the linear equations describing the quasi-linear set in question.