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

Point searching in real singular complete intersection varieties: Algorithms of intrinsic complexity

Abstract. Let X1, . . .,Xn be indeterminates over Q and let X := (X1, . . . ,Xn). Let F1, . . . ,Fp be a regular sequence of polynomials in Q[X] of degreeat most d such that for each 1 ≤ k ≤ p the ideal (F1, . . . , Fk) is radical.Suppose that the variables X1, . . .,Xn are in generic position with respect toF1, . . . ,Fp. Further, suppose that the polynomials are given by an essentiallydivision-free circuit β in Q[X] of size L and non-scalar depth .We present a family of algorithms Πi and invariants δi of F1, . . . ,Fp, 1 ≤i ≤ n − p, such that Πi produces on input β a smooth algebraic sample pointfor each connected component of {x ∈ Rn | F1(x) = ・ ・ ・ = Fp(x) = 0} wherethe Jacobian of F1 = 0, . . . , Fp = 0 has generically rank p.The sequential complexity of Πi is of order L(nd)O(1)(min{(nd)cn, δi})2and its non-scalar parallel complexity is of order O(n( + lognd) log δi). Herec > 0 is a suitable universal constant. Thus, the complexity of Πi meetsthe already known worst case bounds. The particular feature of Πi is itspseudo-polynomial and intrinsic complexity character and this entails the bestruntime behavior one can hope for. The algorithm Πi works in the non-uniformdeterministic as well as in the uniform probabilistic complexity model. Wealso exhibit a worst case estimate of order (nn d)O(n) for the invariant δi. Thereader may notice that this bound overestimates the extrinsic complexity ofΠi, which is bounded by (nd)O(n).1.Fil: Bank, Bernd. Universität zu Berlin; AlemaniaFil: Giusti, Marc. École Polytechnique; Francia. Centre National de la Recherche Scientifique; FranciaFil: Heintz, Joos Ulrich. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; Argentina. Universidad de Cantabria. Facultad de Ciencias. Departamento de Matemáticas, Estadística y Computación; Españ

Bipolar varieties and real solving of a singular polynomial equation

We introduce the concept of a bipolar variety of a real algebraic hypersurface. This notion is then used for the design and complexity estimations of a novel type of algorithms that finds algebraic sample points for the connected components of a singular real hypersurface. The complexity of these algorithms is polynomial in the maximal geometric degree of the bipolar varieties of the given hypersurface and in this sense intrinsic.Fil: Bank, Bernd. Universität zu Berlin; AlemaniaFil: Giusti, Marc. Centre National de la Recherche Scientifique; FranciaFil: Heintz, Joos Ulrich. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Pardo, Luis Miguel. Universidad de Cantabria; Españ

On Bézout inequalities for non-homogeneous polynomial ideals

We introduce a “workable” notion of degree for non-homogeneous polynomial ideals and formulate and prove ideal theoretic Bézout inequalities for the sum of two ideals in terms of this notion of degree and the degree of generators.Fil: Hashemi, Amir. Isfahan University of Technology; IránFil: Heintz, Joos Ulrich. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigación en Ciencias de la Computación. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigación en Ciencias de la Computación; ArgentinaFil: Pardo, Luis M.. Universidad de Cantabria. Facultad de Ciencias; EspañaFil: Solernó, Pablo Luis. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentin

Degeneracy Loci and Polynomial Equation Solving

Let (Formula presented.) be a smooth, equidimensional, quasi-affine variety of dimension (Formula presented.) over (Formula presented.), and let (Formula presented.) be a (Formula presented.) matrix of coordinate functions of (Formula presented.), where (Formula presented.). The pair (Formula presented.) determines a vector bundle (Formula presented.) of rank (Formula presented.) over (Formula presented.). We associate with (Formula presented.) a descending chain of degeneracy loci of (Formula presented.) (the generic polar varieties of (Formula presented.) represent a typical example of this situation). The maximal degree of these degeneracy loci constitutes the essential ingredient for the uniform, bounded-error probabilistic pseudo-polynomial-time algorithm that we will design and that solves a series of computational elimination problems that can be formulated in this framework. We describe applications to polynomial equation solving over the reals and to the computation of a generic fiber of a dominant endomorphism of an affine space.Fil: Bank, Bernd. Universität zu Berlin; AlemaniaFil: Giusti, Marc. Laboratoire D'informatique de L'ecole Polytechnique; FranciaFil: Heintz, Joos Ulrich. Universidad de Cantabria; España. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Lecerf, Grégoire. Laboratoire D'informatique de L'ecole Polytechnique; FranciaFil: Matera, Guillermo. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; ArgentinaFil: Solernó, Pablo Luis. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentin