Firstly, I should mention what I mean by codimension. By Whitney, every algebraic set in $\mathbb{R}^n$ has a stratification and codimension of such a set is defined to be the minimal codimension of all the strata.
Now my question is what can we say about the codimension of the intersection of the zero sets of two polynomials $P,Q\in\mathbb{R}[x_1,\ldots,x_n]$? When does the codimension increase upon taking intersection? Geometrically, this is the case when the intersection is transverse. But how do I check transversality in the case of zero sets?
The case I'm particularly interested in has the following form, $P=x_1f_1g_1, Q=x_2f_2g_2$, where $f_i,g_i$ are distinct, irreducible (even over $\mathbb{C}$), homogeneous, monic polynomials, with degree at least $2$ and each variable having degree $\le 1$, in $\mathbb{R}[x_1,x_2,\ldots, x_n]$. Such a polynomial come as the determinant polynomial of some square matrix formed by the variables $x_i$. What can we say about $\text{codim} (P^{-1}(0)\cap Q^{-1}(0))$ (where the zero sets are considered in $\mathbb{R}^n$)? Could we assert that it is at least $2$?
Any help is appreciated.