On The Schinzel–Wójcik Problem Under Hypothesis H

Abstract Given r non-zero rational numbers $a_1, \ldots, a_r$ which are not $\pm1$ , we complete, under Hypothesis H , a characterisation of the Schinzel–Wójcik r -rational tuples (i.e. r -tuples of rational numbers for which the Schinzel–Wójcik problem has an affirmative answer) which satisfy that the sum of the exponents of the positive elements $a_i$ in the representation of $-1$ in terms of the elements $a_i$ in the multiplicative group $\langle a_1,\dots, a_r\rangle\subset \mathbb{Q}^*$ is even whenever $-1 \in \langle a_1,\dots, a_r\rangle.$