Evaluating geometric queries using few arithmetic operations

Let \cp:=(P_1,...,P_s) be a given family of $n$-variate polynomials with integer coefficients and suppose that the degrees and logarithmic heights of these polynomials are bounded by $d$ and $h$, respectively. Suppose furthermore that for each $1\leq i\leq s$ the polynomial $P_i$ can be evaluated using $L$ arithmetic operations (additions, subtractions, multiplications and the constants 0 and 1). Assume that the family \cp is in a suitable sense \emph{generic}. We construct a database $\cal D$, supported by an algebraic computation tree, such that for each $x\in [0,1]^n$ the query for the signs of $P_1(x),...,P_s(x)$ can be answered using h d^{\cO(n^2)} comparisons and $nL$ arithmetic operations between real numbers. The arithmetic-geometric tools developed for the construction of $\cal D$ are then employed to exhibit example classes of systems of $n$ polynomial equations in $n$ unknowns whose consistency may be checked using only few arithmetic operations, admitting however an exponential number of comparisons

Software Engineering and Complexity in Effective Algebraic Geometry

We introduce the notion of a robust parameterized arithmetic circuit for the evaluation of algebraic families of multivariate polynomials. Based on this notion, we present a computation model, adapted to Scientific Computing, which captures all known branching parsimonious symbolic algorithms in effective Algebraic Geometry. We justify this model by arguments from Software Engineering. Finally we exhibit a class of simple elimination problems of effective Algebraic Geometry which require exponential time to be solved by branching parsimonious algorithms of our computation model.Comment: 70 pages. arXiv admin note: substantial text overlap with arXiv:1201.434

Efficient evaluation of specific queries in constraint databases

Let F1,...,FsεR[X1,...,Xn] be polynomials of degree at most d, and suppose that F1,...,F s are represented by a division free arithmetic circuit of non-scalar complexity size L. Let A be the arrangement of Rn defined by F 1,...,Fs. For any point xεRn, we consider the task of determining the signs of the values F1(x),...,F s(x) (sign condition query) and the task of determining the connected component of A to which x belongs (point location query). By an extremely simple reduction to the well-known case where the polynomials F 1,...,Fs are affine linear (i.e., polynomials of degree one), we show first that there exists a database of (possibly enormous) size sO(L+n) which allows the evaluation of the sign condition query using only (Ln)O(1)log(s) arithmetic operations. The key point of this paper is the proof that this upper bound is almost optimal. By the way, we show that the point location query can be evaluated using dO(n)log(s) arithmetic operations. Based on a different argument, analogous complexity upper-bounds are exhibited with respect to the bit-model in case that F 1,...,Fs belong to Z[X1,...,Xn] and satisfy a certain natural genericity condition. Mutatis mutandis our upper-bound results may be applied to the sparse and dense representations of F 1,...,Fs.Fil: Grimson, Rafael. Hasselt University; Bélgica. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria; ArgentinaFil: Heintz, Joos Ulrich. Universidad de Cantabria; España. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; ArgentinaFil: Kuijpers, Bart. Hasselt University; Bélgic

Quiz Games as a model for Information Hiding

We present a general computation model inspired in the notion of information hiding in software engineering. This model has the form of a game which we call quiz game. It allows in a uniform way to prove exponential lower bounds for several complexity problems of elimination theory.Comment: 46 pages, to appear in Journal of Complexit

On the intrinsic complexity of elimination theory

We consider the intrinsic complexity of selected algorithmic problems of classical elimination theory in algebraic geometry. The inputs and outputs of these problems are given by finite sets of polynomials which we represent alternatively in dense forme or by straight line programs. We begin with an overview on the known upper bounds for the sequential and parallel time complexity of these problems and show then that in the most important cases these bounds are tight. Our lower bound results include both the relative and the absolute viewpoint of complexity theory. On one side we give reductions of fundamental questions of elimination theory to NP- and P#- complete problems and on the other side we show that some of these questions may have exponential size outputs. In this way we confirm the intrinsically exponential character of algorithmic problems in elimination theory whatever the type of data structure may be

On the intrinsic complexity of point finding in real singular hypersurfaces

In previous work we designed an efficient procedure that finds an algebraic sample point for each connected component of a smooth real complete intersection variety. This procedure exploits geometric properties of generic polar varieties and its complexity is intrinsic with respect to the problem. In the present paper we introduce a natural construction that allows to tackle the case of a non–smooth real hypersurface by means of a reduction to a smooth complete intersection

On the geometry of polar varieties

We have developed in the past several algorithms with intrinsic complexity bounds for the problem of point finding in real algebraic varieties. Our aim here is to give a comprehensive presentation of the geometrical tools which are necessary to prove the correctness and complexity estimates of these algorithms. Our results form also the geometrical main ingredients for the computational treatment of singular hypersurfaces. In particular, we show the non--emptiness of suitable generic dual polar varieties of (possibly singular) real varieties, show that generic polar varieties may become singular at smooth points of the original variety and exhibit a sufficient criterion when this is not the case. Further, we introduce the new concept of meagerly generic polar varieties and give a degree estimate for them in terms of the degrees of generic polar varieties. The statements are illustrated by examples and a computer experiment