Multidimensional residue calculus over $\mathbb Q$ or $\overline{\mathbb Q}$

Toric geometry (more specifically within the context of products of projective spaces through the concept of Chow or multi-Chow forms or cycles) plays an important role with respect to upper estimates for the arithmetic complexity of geometric cycles (in terms of logarithmic heights such as Ronkin functions evaluated at the origin) in the affine space $\mathbb A_{\mathbb C}^n$ or the projective space $\mathbb P^n_{\mathbb C}$ which are defined over $\mathbb Q$ or over the algebraically closed field of algebraic numbers $\overline {\mathbb Q}$ (W. Stoll, Y. Nesterenko, D. W. Brownawell, P. Philippon, V. Maillot, J. Kollár, M. Sombra, etc. ). An important formula based on simple linear algebra arguments (basically relying on comparison of dimensions) which arose from Oskar Perron's pioneer book Algebra (1927) [Satz 57, S. 129] appears to be today, since the works of A. Płoski (2005), Z. Jelonek (2005), C. d'Andrea - T. Krick - M. Sombra (2013), a major base stone towards the realization of nearly sharp versions of Hilbert's arithmetic nullstellensatz (or membership problem within the complete intersection setting). Transformation Laws inherent to multivariate residue calculus materialize the bridge between Oskar Perron's result and estimates (which sharpness still remains in general conjectural) for the arithmetic complexity of fully developped trace formulae of the Taylor type (such as Bergman-Weil expansions). I will discuss such questions in this lecture and in particular focus on still un-answered ones. Among such open problems, I will insist on the following : could one expect an effective dynamical approach (based on perturbation arguments which arise from the pioneer work of Krasnoyarsk's mathematical school) be a substitute for the somehow un-effective (but nevertheless extremely efficient!) Oskar Perron's theorem (once combined with Transformation Laws)? A large part of this lecture is inspired by my recent joint work (2021) with Martin Sombra (Barcelona) and the finalization of our book project with Alekos Vidras (Nicosia).