Faster One Block Quantifier Elimination for Regular Polynomial Systems of Equations

Abstract

Quantifier elimination over the reals is a central problem in computational real algebraic geometry, polynomial system solving and symbolic computation. Given a semi-algebraic formula (whose atoms are polynomial constraints) with quantifiers on some variables, it consists in computing a logically equivalent formula involving only unquantified variables. When there is no alternation of quantifiers, one has a one block quantifier elimination problem. This paper studies a variant of the one block quantifier elimination in which we compute an almost equivalent formula of the input. We design a new probabilistic efficient algorithm for solving this variant when the input is a system of polynomial equations satisfying some regularity assumptions. When the input is generic, involves $s$ polynomials of degree bounded by $D$ with $n$ quantified variables and $t$ unquantified ones, we prove that this algorithm outputs semi-algebraic formulas of degree bounded by $\mathcal{D}$ using $O\ {\widetilde{~}}\left ((n-s+1)\ 8^{t}\ \mathcal{D}^{3t+2} \binom{t+\mathcal{D}}{t} \right )$ arithmetic operations in the ground field where $\mathcal{D} = 2(n+s)\ D^s(D-1)^{n-s+1}\ \binom{n}{s}$. In practice, it allows us to solve quantifier elimination problems which are out of reach of the state-of-the-art (up to $8$ variables).