The Dual of Quantifier Elimination: Boolean Elimination over C and R

TL;DR

This paper introduces a dual approach to quantifier elimination in algebraic geometry, providing uniform normal forms with minimal quantifiers for Boolean combinations of polynomial equations over complex and real fields.

Contribution

It establishes the existence of simple, uniform normal forms with minimal quantifiers for Boolean combinations of polynomial equalities and inequalities, extending classical quantifier elimination.

Findings

Normal forms with one existential and one universal quantifier over C and R.

No purely existential or universal normal form is possible over C.

Results apply to constructible sets, entire functions, and functional languages over infinite fields.

Abstract

We show that every finite Boolean combination of polynomial equalities and inequalities in C^n admits two uniform normal forms: an $\exists\forall$ form and a $\forall\exists$ form, each using a single polynomial equation. Both forms use only one existentially quantified variable and one universally quantified variable. Optimality results demonstrate that no purely existential or universal normal form is possible over C. These results extend to sets constructible from entire functions, and to quantifier-free formulas in functional languages over infinite fields of characteristic 0. A corollary shows Zilber's conjecture on quasiminimality equivalent to its subcase quantifying a single equation. Over R and Q, similar results hold, including a singly-quantified $\exists$ form for Boolean combinations of equations and inequations, an $\exists$ form for R established by prior methods, and other results for order inequalities parallel to the forms over C. These results provide a dual to classical quantifier elimination: instead of removing quantifiers at the cost of increased Boolean complexity, they remove Boolean structure at the cost of a short, fixed quantifier prefix. The constructions have linear degree bounds and are explicit.