Real root finding for equivariant semi-algebraic 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 $(sn)^{O(dt)}$

Computing the First Betti Numberand Describing the Connected Components of Semi-algebraic Sets

In this paper we describe a singly exponential algorithm for computing the first Betti number of a given semi-algebraic set. Singly exponential algorithms for computing the zero-th Betti number, and the Euler-Poincar\'e characteristic, were known before. No singly exponential algorithm was known for computing any of the individual Betti numbers other than the zero-th one. We also give algorithms for obtaining semi-algebraic descriptions of the semi-algebraically connected components of any given real algebraic or semi-algebraic set in single-exponential time improving on previous results

Computing the First Few Betti Numbers of Semi-algebraic Sets in Single Exponential Time

In this paper we describe an algorithm that takes as input a description of a semi-algebraic set $S \subset \R^k$, defined by a Boolean formula with atoms of the form $P > 0, P < 0, P=0$ for $P \in {\mathcal P} \subset \R[X_1,...,X_k],$ and outputs the first $\ell+1$ Betti numbers of $S$, $b_0(S),...,b_\ell(S).$ The complexity of the algorithm is $(sd)^{k^{O(\ell)}},$ where where s = #({\mathcal P}) and $d = \max_{P\in {\mathcal P}}{\rm deg}(P),$ which is singly exponential in $k$ for $\ell$ any fixed constant. Previously, singly exponential time algorithms were known only for computing the Euler-Poincar\'e characteristic, the zero-th and the first Betti numbers

Computing roadmaps in smooth real algebraic sets

International audienceLet (f1, . . . , fs) be polynomials in Q[X 1 , . . . , Xn ] of degree bounded by D that generate a radical equidimensional ideal of dimension d and let V ⊂ C^n be the locus of their complex zero set which is supposed to be smooth. A roadmap in V ∩ R^n is a real algebraic curve contained in V ∩ Rn which has a non-empty and connected intersection with each connected component of V ∩ R^n . The classical strategy to compute roadmaps is due to J. Canny and leads to algorithms having a complexity within D^O(n^2) arithmetic operations in Q. This strategy is based on computing a polar variety of dimension 1 and a recursion on the studied variety intersected with fibers taken above a critical value of a projection. Thus, it requires computations with real algebraic numbers and introduces singularities at each recursive call. Thus, no efficient implementation of roadmap algorithms have been obtained until now. Our aim is to provide an efficient implementation of the roadmap algorithm. We show how to slightly modify this strategy in order to avoid the use of real algebraic numbers and to deal with smooth algebraic sets at each recursive call in the case where the input variety is smooth. Our complexity is h^d D^O(n) operations in Q where h bounds the number of recursive call in our algorithm. This quantity is related to the geometry of V ∩ R^n and is bounded by D^O(n), thus in worst cases our algorithm has a complexity within D^O(n^2) arithmetic operations. We report on some experiments done with a preliminary implementation of our algorithm

Computing the homology of basic semialgebraic sets in weak exponential time

We describe and analyze an algorithm for computing the homology (Betti numbers and torsion coefficients) of basic semialgebraic sets which works in weak exponential time. That is, out of a set of exponentially small measure in the space of data the cost of the algorithm is exponential in the size of the data. All algorithms previously proposed for this problem have a complexity which is doubly exponential (and this is so for almost all data)