If we have n homogeneous polynomials (over algebraically closed field) $f_1\ldots , f_n$ on variables $x_0, \ldots , x_n$ $$ f_i(x_0, \ldots , x_n) = \sum_{j_0,\ldots , j_n} a_{i, j_0, \ldots , j_n} x_0^{j_0}\ldots x_n^{j_n} $$ then the common condition is the condition when there are finitely many solutions of the system $f_1=\ldots = f_n=0$.

What is the codimension of non-common condition on coefficients $a_{i, j_0, \ldots , j_n}$ ? Is it true that it is at least 2?

The origin of this question is of following: there is an article of Shakirov http://arxiv.org/abs/0807.4539 where he is stating his formula on resultants. There is a Poisson's lemma: $Res(f_0, \ldots , f_n)=C\prod_{i=1}^N f_0(a_i)$ where $a_i$ are the common zeroes of $f_1, \ldots , f_n$ (if $f_1, \ldots , f_n$ are in common condition) and C is a non-zero element of the field. Then he is using this formula to state an equation on logarithms of resultants. But if the non-common condition is of codimension 1 then his proof fails: the element $C$ may be a rational function with zeroes and poles lying inside non-common condition.

I've wrote him and he said that the editor of his article (in Russian journal "Functional analysis") also is in misunderstanding with his proof.

The common-condition means that the factor $\Bbbk [x_0, \ldots , x_n]_m/(f_1, \ldots , f_n)_m$ (where the index $_m$ means that it is a homogeneous part of degree $m$) is of maximal possible dimension when $m$ is sufficiently large. This gives us a system of inequalities (some determinants are non equal to zero). But this inequalities can still have a common factor...

If you don't understand my English I'm very-very sorry :)