Regular solutions of a congruence system (Q1210108)

Let \(k,n,\nu,\lambda_ \nu,q_ \nu\in\mathbb N\) with \(1\leq\nu\leq n\leq k\) and \(n\geq 2\) and let \(q=\text{lcm}(q_ 1,\dots,q_ n)\). Let \(\gamma_ i,\beta_ i\in\mathbb Z\) satisfy \(\gcd(\gamma_ i,q)\mid\beta_ i\), where \(j+1\leq i\leq k\) for some \(j\in\mathbb N\). In this paper the authors obtain bounds and recurrence formulae for the number of solutions \(x_ 1,\dots,x_ k\) of the system of congruences, \[ \sum_{t=1}^ k x_ t^ \nu \equiv\lambda_ \nu \pmod {q_ \nu} \qquad\text{and}\qquad \gamma_ i x_ i\equiv \beta_ i\pmod q. \] The explicit formulae for the bounds are given in the paper but they are too complicated to mention here.