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

We propose a highly efficient method for variable elimination in polynomial ideals over the rational numbers. In addition, we discuss significant shortcuts for radical computation within de Jong’s normalization algorithm. 

Rather than relying on traditional symbolic elimination, our approach utilizes
modular reconstruction over specialized fibers. By identifying and exploiting the equidimensional and saturated structure of the ideals encountered during normalization, we introduce shortcuts that bypass the need for full Noetherian induction in many cases. By employing linear algebra in finite fields, the ``symbolic barrier'' of coefficient swell is systematically bypassed. A theoretical framework ensures that the process terminates correctly for all equidimensional ideals. Benchmarks demonstrate a dramatic acceleration: A system that caused traditional methods in {\sc Singular} to fail after 6 hours of computation is certified by our approach in 0.45 seconds.