I am looking for a suitable notion to discribe the following property for $\{f_i\}$.
Let $R=k[x_1,\ldots,x_n]$, $\{f_i\}$ is a set a some homogenous polynomials of degree $2$ (or more general, of degree $d$) , such that if $$\sum l_i f_i=0$$ for some homogenous polynomials of degree $1$ (or more general, of degree $<d$), then $l_i=0$ for all $i$.
I know that $(f_i)$ being a regular sequence is a suffcient condition, but it is too strong. For example, we can find $\{f_i\}_{i=1,\ldots,n+1}$ in $R=k[x_1,\ldots,x_n]$ with the property above (a example can be found in my previous question, where we take $R=k[x_1,\ldots,x_5]/(x_1+\ldots+x_5)$), but a $n$-dimensional ring will not admit a $(n+1)$-depth regular sequence.