Real root finding for equivariant semi-algebraic systems

Cordian Riener (1) , Mohab Safey El Din (2)

1 UiT - The Arctic University of Norway [Tromsø, Norway]
2 PolSys - Polynomial Systems

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

Domaines: Calcul formel [cs.SC]; Géométrie algébrique [math.AG]

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⟩