  We introduce a new concept of rank - relative rank associated to a filtered
collection of polynomials. When the filtration is trivial our relative rank
coincides with Schmidt rank (also called strength). We also introduce the
notion of relative bias. The main result of the paper is a relation between
these two quantities over finite fields (as a special case we obtain a new
proof of the results in arXiv:1902.09830). This relation allows us to get an
accurate estimate for the number of points on an affine variety given by a
collection of polynomials which is high relative rank (Lemma 3.2). The key
advantage of relative rank is that it allows one to perform an efficient
regularization procedure which is polynomial in the initial number of
polynomials (the regularization process with Schmidt rank is far worse than
tower exponential). The main result allows us to replace Schmidt rank with
relative rank in many key applications in combinatorics, algebraic geometry and
algebra. For example, we prove that any collection of polynomials
$\mathcal{P}=(P_i)_{i=1}^c$ of degrees $\le d$ in a polynomial ring over an
algebraically closed field of characteristic $>d$ is contained in an ideal
$\mathcal{I}(\mathcal{Q})$, generated by a collection $\mathcal{Q}$ of
polynomials of degrees $\le d$ which form a regular sequence, and $\mathcal{Q}$
is of size $\le A c^{A}$, where $A=A(d)$ is independent of the number of
variables.
