Faster One Block Quantifier Elimination for Regular Polynomial Systems of Equations
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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 𝒟 using O ((n-s+1) 8^t 𝒟^3t+2t+𝒟t ) arithmetic operations in the ground field where 𝒟 = 2(n+s) D^s(D-1)^n-s+1 ns. 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).
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