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Real root finding for equivariant semi-algebraic systems

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Mohab Safey El Din

Abstract

Let $R$ be a real closed field. We consider basic semi-algebraic sets defined by $n$-variate equations/inequalities of $s$ symmetric polynomials and an equivariant family of polynomials, all of them of degree bounded by $2d < n$. Such a semi-algebraic set is invariant by the action of the symmetric group. We show that such a set is either empty or it contains a point with at most $2d-1$ distinct coordinates. Combining this geometric result with efficient algorithms for real root finding (based on the critical point method), one can decide the emptiness of basic semi-algebraic sets defined by $s$ polynomials of degree $d$ in time $(sn)^{O(d)}$. This improves the state-of-the-art which is exponential in $n$. When the variables $x_1, \ldots, x_n$ are quantified and the coefficients of the input system depend on parameters $y_1, \ldots, y_t$, one also demonstrates that the corresponding one-block quantifier elimination problem can be solved in time $(sn)^{O(dt)}$.
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Dates and versions

hal-01819106 , version 1 (20-06-2018)

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Cordian Riener, Mohab Safey El Din. Real root finding for equivariant semi-algebraic systems. ISSAC 20018 - 43rd International Symposium on Symbolic and Algebraic Computation, Jul 2018, New-York, United States. ⟨hal-01819106⟩
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