I am very sorry but apparently I am really weak in cohomology flavored questions. I try to reformulate my problem in a very simple and hopefully clear way. This question is related with a problem in hyperplane arrangement theory, too.

Let $d,h\geq1$ be integer numbers and let $F_{1},\ldots,F_{h}$ be elements of the polynomial ring $\mathbb{C}\left[t_{1},\ldots,t_{d}\right].$ For $1\leq j\leq h$ let $X_{j}=\lbrace P\in\mathbb{C}^{d}\mid F_{j}(P)=0\rbrace$ where $F_{j}(P)$ denotes the evaluation of the polynomial $F_{j}$ at the point $P.$ Finally, set $X=\bigcap_{1\leq j\leq d}X_{j}.$

$\textbf{Central hypothesis:}$ Here everything has the classical topology.

Let $H_{k}(X)$ be the singular homology groups of $X.$

$\textbf{Question:}$ Is there any explicit formula which relate $H_{0}(X),$ i.e., the number of connected components of $X,$ with the polynomials $F_{1},\ldots,F_{h}$ or the spaces $X_{j}=\lbrace P\in\mathbb{C}^{d}\mid F_{j}(P)=0 \rbrace$?