Effective Bounds for Singular Series in the Multivariate Bateman Horn Conjecture

We propose an approach to estimating the error in computing the singular series in the multivariate Bateman Horn conjecture, based on a combination of methods from algebraic geometry and analytic number theory.

For general polynomial systems, we establish a uniform estimate in primes for the local factors, from which we derive a universal upper bound for the relative error expressed in terms of a geometric constant depending on the Betti numbers of the projective closures of the hypersurfaces.

For a single polynomial, an explicit bound for this constant is given in terms of the degree and the number of variables, making the result constructive.

In the diagonal case, using Katz exact formula for diagonal cohomologies, we obtain substantially faster convergence; an additional application of the Hardy Littlewood circle method allows us to further refine the estimate.

Numerical examples show that diagonal systems yield an accuracy gain of several orders of magnitude compared with the general case.

Our results provide rigorous quantitative error control and demonstrate that the convergence rate is determined not only by the degree but also by the geometric structure of the polynomial system.