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3
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q-th powers and roots of polynomials
A family of counterexamples is defined as follows:
$r=2$,
$p_1\ge 2, \quad p_2\ge 3, \quad d_1 \perp p_1, \quad d_2 \perp p_2$,
$a=\dfrac{\tan(\pi d_1/p_1)-\tan(\pi d_2/p_2)}{\tan(\pi d_1/p_1)+\t …