Counting the solutions of λ1x1k1+⋯+λtxtkt≡cmodn

Abstract Given a polynomial Q ( x 1 , ⋯ , x t ) = λ 1 x 1 k 1 + ⋯ + λ t x t k t , for every c ∈ Z and n ≥ 2 , we study the number of solutions N J ( Q ; c , n ) of the congruence equation Q ( x 1 , ⋯ , x t ) ≡ c mod n in ( Z / n Z ) t such that x i ∈ ( Z / n Z ) × for i ∈ J ⊆ I = { 1 , ⋯ , t } . We deduce formulas and an algorithm to study N J ( Q ; c , p a ) for p any prime number and a ≥ 1 any integer. As consequences of our main results, we completely solve: the counting problem of Q ( x i ) = ∑ i ∈ I λ i x i for any prime p and any subset J of I ; the counting problem of Q ( x i ) = ∑ i ∈ I λ i x i 2 in the case t = 2 for any p and J , and the case t general for any p and J satisfying min ⁡ { v p ( λ i ) | i ∈ I } = min ⁡ { v p ( λ i ) | i ∈ J } ; the counting problem of Q ( x i ) = ∑ i ∈ I λ i x i k 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 ⁡ { v p ( λ i ) | i ∈ I } = min ⁡ { v p ( λ i ) | i ∈ J } .