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

Small ideals in polynomial rings and applications