Motivated by applications in combinatorial geometry, we consider the following question: Let $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_m)$ be an $m$-partition of a positive integer $n$, $S_i \subseteq \mathbb{C}^{\lambda_i}$ finite sets, and let $S:=S_1 \times S_2 \cdots S_m \subset \mathbb{C}^n$ multigrid defined $S_i$. Suppose $p$ is $n$-variate degree $d$ polynomial. How many zeros does h...

Abstract Persistence of nondegeneracy is a phenomenon that appears in the theory $\overline{\mathbb{Q}}_l$-representations linear group: every irreducible submodule restriction to mirabolic sub-representation nondegenerate representation nondegenerate. This not true anymore general, if we look at modulo $l$ reduction some stable lattice. As Clozel–Harris–Taylor generalization global Ihara’s lem...

We prove a multidimensional version of the Nazarov lemma. The proof is based on an appropriate generalisation regularised system intervals introduced by J. Mashreghi, F.L. Nazarov, and V.P. Khavin.

The purpose of this paper is to introduce a nonlinear scalarisation function for solving a class of implicit vector equilibrium problems. We prove a scalarisation lemma to show the relation between the implicit vector equilibrium problem and the nonlinear scalarisation function. Then we derive some new existence theorems for solutions of implicit vector equilibrium problems, using the scalarisa...

Let G be a k-partite graph with n vertices in parts such that each vertex is adjacent to at least δ∗(G) vertices in each of the other parts. Magyar and Martin [20] proved that for k = 3, if δ∗(G) ≥ 2 3 n and n is sufficiently large, then G contains a K3-factor (a spanning subgraph consisting of n vertex-disjoint copies of K3) except that G is one particular graph. Martin and Szemerédi [21] prov...