Kutasov-Seiberg dualities and cyclotomic polynomials

We classify all Kutasov-Seiberg type dualities in large $N_c$ SQCD with adjoints of rational $R$-charges. This is done by equating the superconformal index of the electric and magnetic theories: the obtained equation has a solution each time some product of cyclotomic polynomials has only positive coefficients. In this way we easily reproduce without any reference to the superpotential or the choice of the equations of motion (classical chiral ring) all the known dualities from the literature, while adding to them a new family with two adjoints with $R$ charges $\frac{2}{2k+1}$ and $\frac{2(k+1)}{2k+1}$ for all integers $k>1$. We argue that these new fixed points could be in their appropriate conformal windows and in some range of the Yukawas involved a low energy limit of the $D_{2k+2}$ fixed point. We try to clarify some issues connected to the difference between classical and quantum chiral ring of this new solution.