ON POLYNOMIALS WHOSE ROOTS HAVE RATIONAL QUOTIENT OF DIFFERENCES

Abstract

We classify all polynomials$P(X)\in \mathbb{Q}[X]$with rational coefficients having the property that the quotient$(\unicode[STIX]{x1D706}_{i}-\unicode[STIX]{x1D706}_{j})/(\unicode[STIX]{x1D706}_{k}-\unicode[STIX]{x1D706}_{\ell })$is a rational number for all quadruples of roots$(\unicode[STIX]{x1D706}_{i},\unicode[STIX]{x1D706}_{j},\unicode[STIX]{x1D706}_{k},\unicode[STIX]{x1D706}_{\ell })$with$\unicode[STIX]{x1D706}_{k}\neq \unicode[STIX]{x1D706}_{\ell }$.