  Let $\mathbf{k}$ be a field which is either finite or algebraically closed
and let $R = \mathbf{k}[x_1,\ldots,x_n].$ We prove that any $g_1,\ldots,g_s\in
R$ homogeneous of positive degrees $\le d$ are contained in an ideal generated
by an $R_t$-sequence of $\le A(d)(s+t)^{B(d)}$ homogeneous polynomials of
degree $\le d,$ subject to some restrictions on the characteristic of
$\mathbf{k}.$ This yields effective bounds for new cases of Ananyan and
Hochster's theorem A in arXiv:1610.09268 on strength and the codimension of the
singular locus. It also implies effective bounds when $d$ equals the
characteristic of $\mathbf{k}$ for Tao and Ziegler's result in arXiv:1101.1469
on rank and $U^d$ Gowers norms of polynomials over finite fields.
