Formalizing Wu-Ritt Method in Lean 4

TL;DR

This paper formalizes the Wu-Ritt characteristic set method for polynomial systems within Lean 4, providing verified algorithms and foundational proofs for certified algebraic computation.

Contribution

It introduces a formalization of Wu-Ritt's method in Lean 4, including core algebraic notions, algorithms, and correctness proofs for polynomial system decomposition.

Findings

Formalization of polynomial initials, orders, and pseudo-division in Lean 4

Verified algorithms for computing characteristic sets and zero decompositions

Proof of termination and correctness of the formalized algorithms

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.