Quantitative Bounds for the U4-Inverse Theorem over Low Characteristic Finite Fields

This paper gives the first quantitative bounds for the inverse theorem for the Gowers U⁴-norm over (Formula Presented) when p = 2;3. We build upon earlier work of Gowers and Milićević who solved the corresponding problem for p ≥ 5. Our proof has two main steps: symmetrization and integration of low-characteristic trilinear forms. We are able to solve the integration problem for all k-linear forms, but the symmetrization problem we are only able to solve for trilinear forms. We pose several open problems about symmetrization of lowcharacteristic k-linear forms whose resolution, combined with recent work of Gowers and Milićević, would give quantitative bounds for the inverse theorem for the GowersU^k+1-norm over (Formula Presented) for all k; p.