Elimination and the Calculation of the Radical in De Jong's Normalization Algorithm

This paper presents a major algorithmic breakthrough in the constructive normalization of reduced Noetherian rings, focusing on the iterative framework established by the author in 1998. While early implementations of the de Jong algorithm were occasionally criticized for their perceived slowness, we demonstrate that this was unjustly attributed to the iterative loop itself. The true bottleneck was the reliance on traditional symbolic elimination theory. By introducing a specialized fiber-based reconstruction method for radical computation, we bypass the "symbolic barrier" and the resulting coefficient explosion entirely. Our approach replaces expensive symbolic variable suppression with fast linear algebra over finite fields and rational reconstruction. Benchmarks on a standard single-core laptop show that our method processes dense multivariate systems in less than half a second, where state-of-the-art symbolic engines fail to produce a result even after six hours of computation. This leap in performance, exceeding a factor of 45,000, confirms that the original de Jong algorithm, when powered by modular fiber logic, remains the most scalable and competitive method for computing the integral closure of Noetherian affine domains.