Let $f(x)\in\mathbb{Z}[x]$ be a nonconstant polynomial. $n, k$ and $c$ integers such that $n\ge 1$ $k\ge 2$. An integer $a$ is called an $f$-exunit in the ring $\mathbb{Z}_n$ of residue classes modulo $n$ if $\gcd(f(a),n)=1$. In this paper, we use principle cross-classification to derive explicit formula for number ${\mathcal N}_{k,f,c}(n)$ solutions $(x_1,...,x_k)$ congruence $x_1+...+x_k\equi...

Choiceless Polynomial Time (CPT) is one of the candidates in the quest for a logic for polynomial time. It is a strict extension of fixed-point logic with counting (FPC) but to date it is unknown whether it expresses all polynomial-time properties of finite structures. We study the CPT-definability of the isomorphism problem for relational structures of bounded colour class size q (for short, q...

The scope of certain well-studied polynomial-time solvable classes of Satisfiability is investigated relative to a polynomial-time solvable class consisting of what we call matched formulas. The class of matched formulas has not been studied in the literature, probably because it seems not to contain many challenging formulas. Yet, we find that, in some sense, the matched formulas are more nume...

We construct low cardinality admissible meshes for polynomials on three classes of planar compact domains: cartesian graph domains, polar graph domains, and domains with piecewise C 2 boundary, that satisfy a Markov polynomial inequality. 1 Planar cartesian and polar graph domains Let K ⊂ R d be a polynomial determining compact domain (i.e., a polynomial vanishing there vanishes everywhere). We...

 Let $ a_0 (omega), a_1 (omega), a_2 (omega), dots, a_n (omega)$ be a sequence of independent random variables defined on a fixed probability space $(Omega, Pr, A)$. There are many known results for the expected number of real zeros of a polynomial $ a_0 (omega) psi_0(x)+ a_1 (omega)psi_1 (x)+, a_2 (omega)psi_2 (x)+...