Commutator blueprints can be seen as blueprints for constructing RGD systems over $\mathbb {F}_2$ with prescribed commutation relations. In this paper, we construct several families of Weyl-invariant commutator blueprints, mostly of universal type. Also applying another result of the author, we obtain new examples of exotic RGD systems of universal type over $\mathbb {F}_2$. In particular, we generalize Tits’ construction of uncountably many trivalent Moufang twin trees to higher rank, we obtain an example of an RGD system of rank $3$ such that the nilpotency degree of the groups $U_w$ is unbounded, and we construct a commutator blueprint of type $(4, 4, 4)$ that is used to answer a question of Tits from the late $1980$s about twin buildings.

We prove that Brinkmann’s problems are decidable for endomorphisms of $F_n\times F_m$: given $(x,y),(z,w)\in F_n\times F_m$ and $\Phi \in \mathrm {End}(F_n\times F_m)$, it is decidable whether there is some $k\in \mathbb {N}$ such that $(x,y)\Phi ^k=(z,w)$ (or $(x,y)\Phi ^k\sim (z,w)$). We also prove decidability of a two-sided version of Brinkmann’s conjugacy problem for injective endomorphisms which, from the work of Logan, yields a solution to the conjugacy problem in ascending HNN-extensions of $F_n\times F_m$. Finally, we study the dynamics of automorphisms of $F_n\times F_m$ at the infinity, proving that that their dynamics at the infinity is asymptotically periodic, as occurs in the free and free-abelian times free cases.

Special Ricci–Hessian equations on Kähler manifolds $(M,g)$, as defined by Maschler [‘Special Kähler–Ricci potentials and Ricci solitons’, Ann. Global Anal. Geom. 34 (2008), 367–380], involve functions $\tau $ on M and state that, for some function $\alpha $ of the real variable $\tau\kern-0.8pt $, the sum of $\alpha \nabla d\tau\kern-0.8pt $ and the Ricci tensor equals a functional multiple of the metric g, while $\alpha \nabla d\tau\kern-0.8pt $ itself is assumed to be nonzero almost everywhere. Three well-known obvious ‘standard’ cases are provided by (non-Einstein) gradient Kähler–Ricci solitons, conformally-Einstein Kähler metrics, and special Kähler–Ricci potentials. We show that, outside of these three cases, such an equation can only occur in complex dimension two and, at generic points, it must then represent one of three types, for which, up to normalizations, $\alpha =2\cot \tau\kern-0.8pt $, $\alpha =2\coth \tau\kern-0.8pt $, or $\alpha =2\tanh \tau\kern-0.8pt $. We also use the Cartan–Kähler theorem to prove that these three types are actually realized in a ‘nonstandard’ way.

We want to compute generic $\mathrm {Ext}$-spaces of twisted polynomial functors in relation to the $\mathrm {Ext}$-spaces of the untwisted ones. We are able to conjecture a direct formula that is suggestively similar to the one known for ungraded functors. Thanks to the study of a spectral sequence, we get to a total computation in low degrees, with remarkable consequences at the level of generic cohomology. Moreover, a partial computation in all degrees proves the inclusion of a big direct summand and the existence of a suggestive universal class.

Let $\{u_n\}_n$ be a nondegenerate linear recurrence sequence of integers with Binet’s formula given by ${u_n= \sum _{i=1}^{m} P_i(n)\alpha _i^n.}$ Assume $\max _i \vert \alpha _i \vert>1$. In 1977, Loxton and Van der Poorten conjectured that for any $\epsilon>0$, there is an effectively computable constant $C(\epsilon )$ such that if $ \vert u_n \vert < (\max _i\{ \vert \alpha _i \vert \})^{n(1-\epsilon )}$, then $n<C(\epsilon )$. Using results of Schmidt and Evertse, a complete noneffective (qualitative) proof of this conjecture was given by Fuchs and Heintze [‘On the growth of linear recurrences in function fields’, Bull. Aust. Math. Soc. 104(1) (2021), 11–20] and, independently, by Karimov et al. [‘The power of positivity’, Proc. LICS 23 (2023), 1–11]. In this paper, we give an effective upper bound for the number of solutions of the inequality $\vert u_n \vert < (\max _i\{ \vert \alpha _i \vert \})^{n(1-\epsilon )}$, thus extending several earlier results by Schmidt, Schlickewei and Van der Poorten.