Estimating the tail of the singular product in the multivariate Bateman-Horn conjecture

This paper investigates the asymptotics of the number of prime values taken by a polynomial in several variables with integer coefficients. Based on probabilistic heuristics and the multidimensional Bateman Horn conjecture, the expected order of growth of this number is derived. The main focus is on the accuracy of computing the singular series. Using methods of algebraic geometry and analytic number theory, estimates for the tail of the singular product are obtained. For the general case of polynomials satisfying smoothness conditions, an upper bound for the relative error is established. For diagonal polynomials, due to the factorization of exponential sums and formula of Katz, an improvement is achieved both in the decay exponent and in the constant. Comparative tables are provided demonstrating the effectiveness of the diagonal case. The results make it possible to control the accuracy of the singular series approximation in practical computations and can be used for further progress towards the proof of the Bateman Horn conjecture for multivariate polynomials.