we survey some recent results on cocharacters of upper triangular matrices. in particular, we deal both with ordinary and graded cocharacter sequence; we list the principal combinatorial results; we show di erent tech-niques in order to solve similar problems.

We give necessary and sufficient conditions for existence of solutions to a general system complex Monge-Amp\`ere equations on Fano horosymmetric manifolds. In particular, we get coupled K\"ahler-Ricci solitons, Mabuchi metrics twisted K\"ahler-Einstein in terms combinatorial data the manifold.

We present a restricted variable generalization of Warning’s Second Theorem (a result giving a lower bound on the number of solutions of a low degree polynomial system over a finite field, assuming one solution exists). This is analogous to Schauz-Brink’s restricted variable generalization of Chevalley’s Theorem (a result giving conditions for a low degree polynomial system not to have exactly ...