classical mechanics - Holonomic constraints and degrees of freedom

Wikipedia and other sources define holonomic constraints as a function

$$ f(\vec{r}_1, \ldots, \vec{r}_N, t) \equiv 0, $$

and says the number of degrees of freedom in a system is reduced by the number of independent holonomic constraints.

I could take multiple such constraints $f_1, \ldots, f_m$ and formulate them as single one that is fulfilled if and only if all $f_i$ are fulfilled:

$$ f = \sum_{i=1}^{m}{\lvert f_i \rvert}. $$

This combined $f$ would obviously reduce the number of degrees of freedom by $m$ instead of $1$.

Alternatively, to avoid the absolute value, I could use a sum of squares

$$ f = \sum_{i=1}^{m} f_i^2 $$

instead. Where is my error in reasoning?