The resultant of developed systems of Laurent polynomials

Let $R_\Delta(f_1,\dots,f_{n+1})$ be the $\Delta$-resultant (defined in the paper) of $(n+1)$-tuple of Laurent polynomials. We provide an algorithm for computing $R_\Delta$ assuming that an $n$-tuple $(f_2,\dots,f_{n+1})$ is developed. We provide a relation between the product of $f_1$ over roots of $f_2=\dots=f_{n+1}=0$ in $(\mathbf C^*)^n$ and the product of $f_2$ over roots of $f_1=f_3=\dots=f_{n+1}=0$ in $(\mathbf C^*)^n$ assuming that the $n$-tuple $(f_1f_2,f_3,\dots,f_{n+1})$ is developed. If all $n$-tuples contained in $(f_1,\dots,f_{n+1})$ are developed we provide a signed version of Poisson formula for $R_\Delta$. In our proofs we use topological arguments and topological version of the Parshin reciprocity laws.