Formalizing Wu-Ritt Method in Lean 4
Yuxuan Xiao, Hao Shen, Junyu Guo, Dingkang Wang, Lihong Zhi
math.AC
Apr 16, 2026 · v1
TL;DR
Formalizes the Wu-Ritt characteristic-set method in Lean 4 with termination/correctness proofs.
Abstract
We formalize the Wu-Ritt characteristic set method for the triangular decomposition of polynomial systems in the Lean 4 theorem prover. Our development includes the core algebraic notions of the method, such as polynomial initials, orders, pseudo-division, pseudo-remainders with respect to a polynomial or a triangular set, and standard and weak ascending sets. On this basis, we formalize algorithms for computing basic sets, characteristic sets, and zero decompositions, and prove their termination and correctness. In particular, we formalize the well-ordering principle relating a polynomial system to its characteristic set and verify that zero decomposition expresses the zero set of the original system as a union of zero sets of triangular sets away from the zeros of the corresponding initials. This work provides a machine-checked verification of Wu-Ritt's method in Lean 4 and establishes a foundation for certified polynomial system solving and geometric theorem proving.
Problem
The Wu-Ritt characteristic set method for triangular decomposition of polynomial systems lacks machine-checked verification, despite its importance for certified polynomial system solving and geometric theorem proving.
Approach
The authors formalize the Wu-Ritt method in Lean 4, developing the core algebraic notions including polynomial initials, orders, pseudo-division, pseudo-remainders with respect to a polynomial or a triangular set, and standard and weak ascending sets. They formalize algorithms for computing basic sets, characteristic sets, and zero decompositions, and prove their termination and correctness in Lean 4.
Results
The formalization verifies the well-ordering principle relating a polynomial system to its characteristic set and confirms that zero decomposition expresses the zero set of the original system as a union of zero sets of triangular sets away from the zeros of the corresponding initials. This provides the first machine-checked verification of Wu-Ritt's method.
