Double-exponential growth of the number of vectors of solutions of polynomial systems

In [4] it was proved an upper bound $d^{O\left(\left(\smallmatrix n+d\\n\endsmallmatrix\right)\right)}$ on the number of vectors of multiplicities of the solutions of systems of the form $g_1=\ldots=g_n=0$ (provided, it has a finite number of solutions) of polynomials $g_1,\dots,g_n\in F[X_1,\dots,X_n]$ with degrees $\deg g_i\le d$ (the field $F$ is algebraically closed). In the present paper it is shown that this bound is close in order to the exact one. In particular, in case $d=n$ the construction provides a double-exponential (in $n$) number of vectors of multiplicities.