Let $\left\{U_n\right\}=\left\{U_n(P,Q)\right\}$ and $\left\{V_n\right\}=\left\{V_n(P,Q)\right\}$ be the Lucas sequences of the first and second kind respectively at the parameters $P\ge 1$ and $Q\in \{-1,1\}$. In this paper, we provide a technique for characterizing the solutions of the so-called Bartz-Marlewski equation $$x^2-3xy+y^2+x=0,$$ where $(x,y)=(U_i,U_j)$ or $(V_i,V_j)$ with $i$, $j\ge 1$. Then, the procedure of this technique is applied to completely resolve this equation with certain values of such parameters.

We count integer points on varieties given by bihomogeneous equations using the Hardy-Littlewood method. The main novelty lies in using the structure of bihomogeneous equations to obtain asymptotics in generically fewer variables than would be necessary in using the standard approach for homogeneous varieties. Also, we consider counting functions where not all the variables have to lie in intervals of the same size, which arises as a natural question in the setting of bihomogeneous varieties.

This paper deals with conditions of compatibility of a system of copulas and with bounds of general Fréchet classes. Algebraic search for the bounds is interpreted as a solution to a linear system of Diophantine equations. Classical analytical specification of the bounds is described.