Complexity of Computing the Local Dimension of a Semialgebraic Set

AbstractThe paper describes several algorithms related to a problem of computing the local dimension of a semialgebraic set. Let a semialgebraic set V be defined by a system of k inequalities of the formf≥ 0 with f∈R [ X1,⋯ ,Xn ], deg(f) <d , andx∈V . An algorithm is constructed for computing the dimension of the Zariski tangent space to V at x in time (kd)O(n). Let x belong to a stratum of codimension lxin V with respect to a smooth stratification ofV . Another algorithm computes the local dimension dimx(V) with the complexity (k(lx+ 1)d)O(lx2n). Ifl=maxx∈Vlx, and for every connected component the local dimension is the same at each point, then the algorithm computes the dimension of every connected component with complexity (k(l+ 1)d)O(l2n). If V is a real algebraic variety defined by a system of equations, then the complexity of the algorithm is less thankdO(l2n) , and the algorithm also finds the dimension of the tangent space to V at x in time kdO(n). Whenl is fixed, like in the case of a smooth V , the complexity bounds for computing the local dimension are (kd)O(n)andkdO(n) respectively. A third algorithm finds the singular locus ofV in time (kd)O(n2)

Computing the dimension of real algebraic sets

Let $V$ be the set of real common solutions to $F = (f_1, \ldots, f_s)$ in $\mathbb{R}[x_1, \ldots, x_n]$ and $D$ be the maximum total degree of the $f_i$'s. We design an algorithm which on input $F$ computes the dimension of $V$. Letting $L$ be the evaluation complexity of $F$ and $s=1$, it runs using $O^\sim \big (L D^{n(d+3)+1}\big )$ arithmetic operations in $\mathbb{Q}$ and at most $D^{n(d+1)}$ isolations of real roots of polynomials of degree at most $D^n$. Our algorithm depends on the real geometry of $V$; its practical behavior is more governed by the number of topology changes in the fibers of some well-chosen maps. Hence, the above worst-case bounds are rarely reached in practice, the factor $D^{nd}$ being in general much lower on practical examples. We report on an implementation showing its ability to solve problems which were out of reach of the state-of-the-art implementations.Comment: v2: title chang

Computing the Real Isolated Points of an Algebraic Hypersurface

Let $\mathbb{R}$ be the field of real numbers. We consider the problem of computing the real isolated points of a real algebraic set in $\mathbb{R}^n$ given as the vanishing set of a polynomial system. This problem plays an important role for studying rigidity properties of mechanism in material designs. In this paper, we design an algorithm which solves this problem. It is based on the computations of critical points as well as roadmaps for answering connectivity queries in real algebraic sets. This leads to a probabilistic algorithm of complexity $(nd)^{O(n\log(n))}$ for computing the real isolated points of real algebraic hypersurfaces of degree $d$. It allows us to solve in practice instances which are out of reach of the state-of-the-art.Comment: Conference paper ISSAC 202