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 alternate of quantifier , one has a one block quantifier elimination problem.  We design a new practically efficient algorithm for solving one block quantifier elimination problems when the input semi-algebraic formula is a system of polynomial equations satisfying some mild assumptionssuch as transversality . 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\ {~}\left ( n \ 8^{t}\ D^{3t+2} \ \binom{t+\mathcal{D}}{t} \right ) arithmetic operations in the ground field where \mathcal{D} = n \ 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).
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\ {~}\left ( (n-s+1) \ 8^{t}\ 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).
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alternate of quantifier
alternation of quantifiers
A
This paper studies a variant of the one block quantifier elimination in which we compute an almost equivalent formula of the input.
R
practically
probabilistic
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one block quantifier elimination problems
this variant
D
semi-algebraic formula
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mild assumptionssuch as transversality
regularity assumptions
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n
(n-s+1)
D
\
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n
2(n+s)