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We find the free field construction of the basic $W$-current and screeningcurrents for the deformed $W$-superalgebra ${\cal W}_{q,t}\bigl(A(M,N)\bigr)$associated with Lie superalgebra of type $A(M,N)$. Using this free fieldconstruction, we introduce the higher $W$-currents and obtain a closed set ofquadratic relations among them. These relations are independent of the choiceof Dynkin-diagrams for the Lie superalgebra $A(M,N)$, though the screeningcurrents are not. This allows us to define ${\cal W}_{q,t}\bigl(A(M,N)\bigr)$by generators and relations.

Quadratic relations of the deformed $W$-superalgebra ${\cal W}_{q, t}\bigl(A(M,N)\bigr)$