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We consider rational integrable supersymmetric gl(m|n) spin chains in thedefining representation and prove the isomorphism between a commutative algebraof conserved charges (the Bethe algebra) and a polynomial ring (the Wronskianalgebra) defined by functional relations between Baxter Q-functions that wecall Wronskian Bethe equations. These equations, in contrast to standard nestedBethe equations, admit only physical solutions for any value of inhomogeneitiesand furthermore we prove that the algebraic number of solutions to theseequations is equal to the dimension of the spin chain Hilbert space (modulorelevant symmetries).  Both twisted and twist-less periodic boundary conditions are considered, theisomorphism statement uses, as a sufficient condition, that the spin chaininhomogeneities $\theta_\ell$, $\ell=1,\ldots,L$ satisfy$\theta_\ell+\hbar\neq\theta_{\ell'}$ for $\ell<\ell'$. Counting of solutionsis done in two independent ways: by computing a character of the Wronskianalgebra and by explicitly solving the Bethe equations in certain scalingregimes supplemented with a proof that the algebraic number of solutions is thesame for any value of $\theta_\ell$. In particular, we consider the regime$\theta_{\ell+1}/\theta_{\ell}\gg 1$ for the twist-less chain where we succeedto provide explicit solutions and their systematic labelling with standardYoung tableaux.

Completeness of Wronskian Bethe equations for rational gl(m|n) spin chains