Relative Rank and Regularization

We introduce a new notion of rank - relative rank associated to a filtered collection of polynomials. When the filtration consists of one set the notion coincides with the Schmidt rank (also called strength). We also introduce the notion of relative bias. We prove a relation between these two notions over finite fields. This allows us to perform a regularization procedure in a number of steps that is polynomial in the initial number of polynomials. As an application we prove that any collection of homogeneous 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 homogeneous 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.