Counting and computing regions of DD-decomposition: algebro-geometric approach

New methods for $D$-decomposition analysis are presented. They are based on topology of real algebraic varieties and computational real algebraic geometry. The estimate of number of root invariant regions for polynomial parametric families of polynomial and matrices is given. For the case of two parametric family more sharp estimate is proven. Theoretic results are supported by various numerical simulations that show higher precision of presented methods with respect to traditional ones. The presented methods are inherently global and could be applied for studying $D$-decomposition for the space of parameters as a whole instead of some prescribed regions. For symbolic computations the Maple v.14 software and its package RegularChains are used.Comment: 16 pages, 8 figure

Polynomial-Time Amoeba Neighborhood Membership and Faster Localized Solving

We derive efficient algorithms for coarse approximation of algebraic hypersurfaces, useful for estimating the distance between an input polynomial zero set and a given query point. Our methods work best on sparse polynomials of high degree (in any number of variables) but are nevertheless completely general. The underlying ideas, which we take the time to describe in an elementary way, come from tropical geometry. We thus reduce a hard algebraic problem to high-precision linear optimization, proving new upper and lower complexity estimates along the way.Comment: 15 pages, 9 figures. Submitted to a conference proceeding

Computation of the singularities of parametric plane curves

Given an algebraic plane curve C defined by a rational parametrization P(t), we present formulae for the computation of the degree of C, and the multiplicity of a point. Using the results presented in [Sendra, J.R., Winkler, F., 2001. Tracing index of rational curve parametrizations. Computer Aided Geometric Design 18 (8), 771–795], the formulae simply involve the computation of the degree of a rational function directly determined from P(t). Furthermore, we provide a method for computing the singularities of C and analyzing the non-ordinary ones without knowing its defining polynomial. This approach generalizes the results in [Abhyankar, S., 1990. Algebraic geometry for scientists and engineers. In: Mathematical Surveys and Monographs, vol. 35. American Mathematical Society; van den Essen, A., Yu, J.-T., 1997. The D-resultants, singularities and the degree of unfaithfulness. Proceedings of the American Mathematical Society 25, 689–695; Gutierrez, J., Rubio, R., Yu, J.-T., 2002. D-Resultant for rational functions. Proceedings of the American Mathematical Society 130 (8), 2237–2246] and [Park, H., 2002. Effective computation of singularities of parametric affine curves. Journal of Pure and Applied Algebra 173, 49–58].Ministerio de Educación y CienciaComunidad de MadridUniversidad de Alcal

Resultants and subresultants of p-adic polynomials

We address the problem of the stability of the computations of resultants and subresultants of polynomials defined over complete discrete valuation rings (e.g. Zp or k[[t]] where k is a field). We prove that Euclide-like algorithms are highly unstable on average and we explain, in many cases, how one can stabilize them without sacrifying the complexity. On the way, we completely determine the distribution of the valuation of the principal subresultants of two random monic p-adic polynomials having the same degree

An in depth analysis, via resultants, of the singularities of a parametric curve

Let C be an algebraic space curve defined by a rational parametrization P(t)∈K(t)ℓ, ℓ≥2. In this paper, we consider the T-function, T(s), which is a polynomial constructed from P(t) by means of a univariate resultant, and we show that T(s) contains essential information concerning the singularities of C. More precisely, we prove that T(s)=∏i=1nHPi(s), where Pi, i=1,…,n, are the (ordinary and non-ordinary) singularities of C and HPi, i=1,…,n, are polynomials, each of them associated to a singularity, whose factors are the fiber functions of those singularities as well as those other belonging to their corresponding neighborhoods. That is, HQ(s)=HQ(s)m−1∏j=1kHQj(s)mj−1, where Q is an m-fold point, Qj,j=1,…,k, are the neighboring singularities of Q, and mj,j=1,…,k, are their corresponding multiplicities (HP denotes the fiber function of P). Thus, by just analyzing the factorization of T, we can obtain all the singularities (ordinary and non-ordinary) as well as interesting data relative to each of them, like its multiplicity, character, fiber or number of associated tangents. Furthermore, in the case of non-ordinary singularities, we can easily get the corresponding number of local branches and delta invariant.Ministerio de Ciencia, Innovación y Universidade

Oscillatory integral operators with homogeneous polynomial phases in several variables

We obtain $L^2$ decay estimates in $\lambda$ for oscillatory integral operators whose phase functions are homogeneous polynomials of degree m and satisfy various genericity assumptions. The decay rates obtained are optimal in the case of (2+2)--dimensions for any m while, in higher dimensions, the result is sharp for m sufficiently large. The proof for large $m$ follows from essentially algebraic considerations. For cubics in (2+2)--dimensions, the proof involves decomposing the operator near the conic zero variety of the determinant of the Hessian of the phase function, using an elaboration of the general approach of Phong and Stein [1994].Comment: 39 pages, 2 figures; minor corrections; to appear in Journal of Functional Analysi