Obstructions and dualities for matroid depth parameters

Contraction${}^{\ast }$-depth is considered to be one of the analogues of graph tree-depth in the matroid setting. In this paper, we investigate structural properties of contraction${}^{\ast }$-depth of matroids representable over finite fields and rationals. In particular, we prove that the obstructions for contraction${}^{\ast }$-depth for these classes of matroids are bounded in size. From this we derive analogous results for related notions of contraction-depth and deletion-depth. Moreover, we define a dual notion to contraction${}^{\ast }$-depth, named deletion${}^{\ast }$-depth, for $\mathbb{F}$-representable matroids, and by duality extend our results from contraction${}^{\ast }$-depth to this notion.