  We examine multidimensional polynomial progressions involving linearly
independent polynomials in finite fields, proving power saving bounds for sets
lacking such configurations. This jointly generalises earlier results of Peluse
(for the single dimensional case) and the author (for distinct degree
polynomials). In contrast to the cases studied in the aforementioned two
papers, a usual PET induction argument does not give Gowers norm control over
multidimensional progressions that involve polynomials of the same degrees. The
main challenge is therefore to obtain Gowers norm control, and we accomplish
this for all multidimensional polynomial progressions with pairwise independent
polynomials. The key inputs are: (1) a quantitative version of a PET induction
scheme developed in ergodic theory by Donoso, Koutsogiannis, Ferr\'e-Moragues
and Sun, (2) a quantitative concatenation result for Gowers box norms in
arbitrary finite abelian groups, motivated by earlier results of Tao, Ziegler,
Peluse and Prendiville; (3) an adaptation to combinatorics of the box norm
smoothing technique, recently developed in the ergodic setting by the author
and Frantzikinakis; and (4) a new version of the multidimensional degree
lowering argument.
