We consider the termination problem for triangular weakly non-linear loops (twn-loops) over some ring S like Z, Q, or R. Essentially, the guard of such a loop is an arbitrary Boolean formula over (possibly non-linear) polynomial inequations, and the body is a single assignment (x₁,…,x_d)⟵(c₁⋅x₁+p₁,…,c_d⋅x_d+p_d) where each x_i is a variable, c_i∈S, and each p_i is a (possibly non-linear) polynomial over S and the variables x_i+1,…,x_d. We present a reduction from the question of termination to the existential fragment of the first-order theory of S and R. For loops over R, our reduction entails decidability of termination. For loops over Z and Q, it proves semi-decidability of non-termination. Furthermore, we present a transformation to convert certain non-twn-loops into twn-form. Then the original loop terminates iff the transformed loop terminates over a specific subset of R, which can also be checked via our reduction. This transformation also allows us to prove tight complexity bounds for the termination problem for two important classes of loops which can always be transformed into twn-loops.