Counting the solutions of λ1x1k1+⋯+λtxtkt≡cmodn

Given a polynomial Q(x1,⋯,xt)=λ1x1k1+⋯+λtxtkt, for every c∈Z and n≥2, we study the number of solutions NJ(Q;c,n) of the congruence equation Q(x1,⋯,xt)≡cmodn in (Z/nZ)t such that xi∈(Z/nZ)× for i∈J⊆I={1,⋯,t}. We deduce formulas and an algorithm to study NJ(Q;c,pa) for p any prime number and a≥1 any integer. As consequences of our main results, we completely solve: the counting problem of Q(xi)=∑i∈Iλixi for any prime p and any subset J of I; the counting problem of Q(xi)=∑i∈Iλixi2 in the case t=2 for any p and J, and the case t general for any p and J satisfying min⁡{vp(λi)|i∈I}=min⁡{vp(λi)|i∈J}; the counting problem of Q(xi)=∑i∈Iλixik in the case t=2 for any p∤k and any J, and in the case t general for any p∤k and J satisfying min⁡{vp(λi)|i∈I}=min⁡{vp(λi)|i∈J}.