🤖 AI Summary
This paper addresses the quantifier normalization problem for Boolean combinations of polynomial equalities and inequalities over ℂ and ℝ. We introduce the notion of “Boolean elimination” and construct a unified normal form containing exactly one existential and one universal quantifier (∃∀ or ∀∃), which equivalently reduces any such Boolean combination to a single polynomial equation while preserving linear degree bounds. We prove that purely existential or purely universal normal forms are impossible over ℂ, thereby establishing the theoretical optimality of this two-quantifier structure. Our approach integrates tools from algebraic number theory, model theory, and polynomial system analysis, and we extend the results to ℚ. Compared to classical quantifier elimination, this paradigm drastically simplifies the Boolean structure while achieving an optimal trade-off between quantifier complexity and expressive power.
📝 Abstract
We show that every finite Boolean combination of polynomial equalities and inequalities in C^n admits two uniform normal forms: an $existsforall$ form and a $forallexists$ form, each using a single polynomial equation. Both forms have one existentially quantified variable and one universally quantified variable; regardless of the complexity of the original formula, no longer quantifier blocks are needed. The constructions are explicit and have linear degree bounds. Optimality results demonstrate that no purely existential or universal normal form is possible over C.
Over R, similar normal forms exist, including a singly-quantified $exists$ form for Boolean combinations of equations and inequations, and $exists^d$ and $forallexists$ forms for Boolean combinations involving order inequalities. Prior results establish the existence of a $exists$ normal form for R by other methods. Finally, similar forms exist over Q as well.
These results may be viewed as 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.