Formal proofs in real algebraic geometry: from ordered fields to quantifier elimination

This paper describes a formalization of discrete real closed fields in the Coq proof assistant. This abstract structure captures for instance the theory of real algebraic numbers, a decidable subset of real numbers with good algorithmic properties. The theory of real algebraic numbers and more generally of semi-algebraic varieties is at the core of a number of effective methods in real analysis, including decision procedures for non linear arithmetic or optimization methods for real valued functions. After defining an abstract structure of discrete real closed field and the elementary theory of real roots of polynomials, we describe the formalization of an algebraic proof of quantifier elimination based on pseudo-remainder sequences following the standard computer algebra literature on the topic. This formalization covers a large part of the theory which underlies the efficient algorithms implemented in practice in computer algebra. The success of this work paves the way for formal certification of these efficient methods.

💡 Research Summary

The paper presents a comprehensive formalization of discrete real closed fields within the Coq proof assistant and uses this foundation to mechanize an algebraic proof of quantifier elimination based on pseudo‑remainder sequences (PRS). The authors begin by defining an abstract structure called a “discrete real closed field,” which captures the essential properties of real closed fields—most notably the decidability of sign and root existence for polynomials—while restricting coefficients and operations to a finite, algorithmically manageable representation. This abstraction makes it possible to encode polynomial arithmetic, Euclidean division, and sign reasoning directly in Coq’s constructive type theory.

A central technical contribution is the formal development of PRS. Unlike the classical Euclidean algorithm, PRS works with scaled remainders that preserve the discreteness of coefficients, preventing the exponential blow‑up that typically hampers formal verification of algebraic algorithms. The authors define PRS, subresultants, and resultants in Coq, prove their basic algebraic properties, and show how PRS can be used to decide the existence of real roots for a pair of polynomials.

Building on PRS, the paper formalizes the sign‑variation theorem and Sturm’s theorem for real closed fields. These results give a precise relationship between the number of sign changes of a polynomial sequence on an interval and the number of real roots in that interval. The authors implement Coq tactics that automatically compute sign variations using PRS, thereby providing a certified procedure for counting real roots and for determining the sign of a polynomial on any semi‑algebraic set.

With these tools in place, the authors turn to quantifier elimination (QE) for the first‑order theory of real closed fields. Traditional QE proofs rely on cylindrical algebraic decomposition (CAD), which is computationally heavy and difficult to formalize. Instead, the paper follows the classical algebraic approach found in computer algebra literature: it iteratively eliminates quantified variables by constructing resultant‑based projection polynomials, applying PRS to maintain control over coefficient growth, and using sign‑variation information to decide existential or universal conditions. Each elimination step is proved correct in Coq, showing that the resulting quantifier‑free formula is logically equivalent to the original.

The implementation is demonstrated on concrete examples, such as deciding the existence of solutions to systems of polynomial equations and inequalities involving real algebraic numbers. The Coq scripts combine built‑in tactics (ring, field, polynomial) with custom PRS and sign‑variation tactics, achieving a high degree of automation while preserving full formal verification.

Finally, the authors discuss the broader impact of their work. By providing a certified core for real algebraic geometry—covering discrete real closed fields, root‑finding via PRS, sign reasoning, and quantifier elimination—the paper paves the way for formally verified decision procedures for non‑linear real arithmetic, certified optimization algorithms, and trustworthy computer‑algebra systems. The success of this formalization demonstrates that sophisticated algebraic algorithms can be brought under the umbrella of interactive theorem proving, offering a new level of reliability for applications that depend on real‑algebraic computation.