Real root finding for equivariant semi-algebraic systems

TL;DR

This paper introduces a geometric property of symmetric semi-algebraic sets that enables more efficient algorithms for real root finding and quantifier elimination, significantly improving computational complexity.

Contribution

It proves a new geometric property of symmetric semi-algebraic sets and develops algorithms with subexponential complexity for root finding and quantifier elimination.

Findings

Semi-algebraic sets contain points with limited distinct coordinates.

Deciding emptiness can be done in time (sn)^{O(d)}.

Quantifier elimination is feasible in time (sn)^{O(dt)}.

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 $(sn)^{O(dt)}$.