Consider $(f\_1,\dots,f\_n), (g\_1,\dots,g\_n)\in \mathbb{C}[z\_1,\dots,z\_n]\ $ such that:


i) $\{f\_1=\dots=f\_n=0\}= \{g\_1=\dots=g\_n=0\}=\{0\}\in \mathbb{C}^n\ $ and 


ii) $f\_1g\_1+\dots+f\_ng\_n\equiv0$.


What is the relation between


$
\displaystyle\dim \frac{\mathbb{C}[z\_1,\dots,z\_n]}{(f\_1,\dots,f\_n)}
$
and
$
\displaystyle\dim \frac{\mathbb{C}[z\_1,\dots,z\_n]}{(g\_1,\dots,g\_n)}
$ ?



Relation between intersection multiplicities
Put $S=\mathbb{C}[z\_1,\ldots ,z\_n]$. Since $V(f\_1,\ldots ,f\_n)=\{0\} $, $(f\_1,\ldots ,f\_n)$ is a $S$-regular sequence, so the (partial) Koszul complex
$$\wedge^2S^n\xrightarrow{\ u\ } S^n\xrightarrow{(f\_1,\ldots ,f\_n)} S$$is exact, with $u(e\_i\wedge e\_j)=f\_ie\_j-f\_je\_i$. This implies $(g\_1,\ldots ,g\_n)\subset (f\_1,\ldots ,f\_n)$. Since the $f\_i$ and $g\_j$ play a symmetric role, we have in fact $\ (g\_1,\ldots ,g\_n)=(f\_1,\ldots ,f\_n)$.


