Rigid continuation paths I. Quasilinear average complexity for solving polynomial systems

How many operations do we need on the average to compute an approximate root of a random Gaussian polynomial system? Beyond Smale's 17th problem that asked whether a polynomial bound is possible, we prove a quasi-optimal bound $\text{(input size)}^{1+o(1)}$. This improves upon the previously known $\text{(input size)}^{\frac32 +o(1)}$ bound. The new algorithm relies on numerical continuation along \emph{rigid continuation paths}. The central idea is to consider rigid motions of the equations rather than line segments in the linear space of all polynomial systems. This leads to a better average condition number and allows for bigger steps. We show that on the average, we can compute one approximate root of a random Gaussian polynomial system of~$n$ equations of degree at most $D$ in $n+1$ homogeneous variables with $O(n^5 D^2)$ continuation steps. This is a decisive improvement over previous bounds that prove no better than $\sqrt{2}^{\min(n, D)}$ continuation steps on the average

Heat kernel regularization of the effective action for stochastic reaction-diffusion equations

The presence of fluctuations and non-linear interactions can lead to scale dependence in the parameters appearing in stochastic differential equations. Stochastic dynamics can be formulated in terms of functional integrals. In this paper we apply the heat kernel method to study the short distance renormalizability of a stochastic (polynomial) reaction-diffusion equation with real additive noise. We calculate the one-loop {\emph{effective action}} and its ultraviolet scale dependent divergences. We show that for white noise a polynomial reaction-diffusion equation is one-loop {\emph{finite}} in $d=0$ and $d=1$, and is one-loop renormalizable in $d=2$ and $d=3$ space dimensions. We obtain the one-loop renormalization group equations and find they run with scale only in $d=2$.Comment: 21 pages, uses ReV-TeX 3.

An excursion from enumerative goemetry to solving systems of polynomial equations with Macaulay 2

Solving a system of polynomial equations is a ubiquitous problem in the applications of mathematics. Until recently, it has been hopeless to find explicit solutions to such systems, and mathematics has instead developed deep and powerful theories about the solutions to polynomial equations. Enumerative Geometry is concerned with counting the number of solutions when the polynomials come from a geometric situation and Intersection Theory gives methods to accomplish the enumeration. We use Macaulay 2 to investigate some problems from enumerative geometry, illustrating some applications of symbolic computation to this important problem of solving systems of polynomial equations. Besides enumerating solutions to the resulting polynomial systems, which include overdetermined, deficient, and improper systems, we address the important question of real solutions to these geometric problems. The text contains evaluated Macaulay 2 code to illuminate the discussion. This is a chapter in the forthcoming book "Computations in Algebraic Geometry with Macaulay 2", edited by D. Eisenbud, D. Grayson, M. Stillman, and B. Sturmfels. While this chapter is largely expository, the results in the last section concerning lines tangent to quadrics are new.Comment: LaTeX 2e, 22 pages, 1 .eps figure. Source file (.tar.gz) includes Macaulay 2 code in article, as well as Macaulay 2 package realroots.m2 Macaulay 2 available at http://www.math.uiuc.edu/Macaulay2 Revised with improved exposition, references updated, Macaulay 2 code rewritten and commente

Gravitation with superposed Gauss--Bonnet terms in higher dimensions: Black hole metrics and maximal extensions

Our starting point is an iterative construction suited to combinatorics in arbitarary dimensions d, of totally anisymmetrised p-Riemann 2p-forms (2p\le d) generalising the (1-)Riemann curvature 2-forms. Superposition of p-Ricci scalars obtained from the p-Riemann forms defines the maximally Gauss--Bonnet extended gravitational Lagrangian. Metrics, spherically symmetric in the (d-1) space dimensions are constructed for the general case. The problem is directly reduced to solving polynomial equations. For some black hole type metrics the horizons are obtained by solving polynomial equations. Corresponding Kruskal type maximal extensions are obtained explicitly in complete generality, as is also the periodicity of time for Euclidean signature. We show how to include a cosmological constant and a point charge. Possible further developments and applications are indicated.Comment: 13 pages, REVTEX. References and Note Adde

ON NEWTON-RAPHSON METHOD

Recent versions of the well-known Newton-Raphson method for solving algebraic equations are presented. First of these is the method given by J. H. He in 2003. He reduces the problem to solving a second degree polynomial equation. However He’s method is not applicable when this equation has complex roots. In 2008, D. Wei, J. Wu and M. Mei eliminated this deficiency, obtaining a third order polynomial equation, which has always a real root. First of the authors of present paper obtained higher order polynomial equations, which for orders 2 and 3 are reduced to equations given by He and respectively by Wei-Wu-Mei, with much improved form. In this paper, we present these methods. An example is given.newton-raphson

Exact polynomial solutions of second order differential equations and their applications

We find all polynomials $Z(z)$ such that the differential equation $${X(z)\frac{d^2}{dz^2}+Y(z)\frac{d}{dz}+Z(z)}S(z)=0,$$ where $X(z), Y(z), Z(z)$ are polynomials of degree at most 4, 3, 2 respectively, has polynomial solutions $S(z)=\prod_{i=1}^n(z-z_i)$ of degree $n$ with distinct roots $z_i$. We derive a set of $n$ algebraic equations which determine these roots. We also find all polynomials $Z(z)$ which give polynomial solutions to the differential equation when the coefficients of X(z) and Y(z) are algebraically dependent. As applications to our general results, we obtain the exact (closed-form) solutions of the Schr\"odinger type differential equations describing: 1) Two Coulombically repelling electrons on a sphere; 2) Schr\"odinger equation from kink stability analysis of $\phi^6$-type field theory; 3) Static perturbations for the non-extremal Reissner-Nordstr\"om solution; 4) Planar Dirac electron in Coulomb and magnetic fields; and 5) O(N) invariant decatic anharmonic oscillator.Comment: LaTex 25 page

A Borel open cover of the Hilbert scheme

Let $p(t)$ be an admissible Hilbert polynomial in \PP^n of degree $d$. The Hilbert scheme \hilb^n_p(t) can be realized as a closed subscheme of a suitable Grassmannian $\mathbb G$, hence it could be globally defined by homogeneous equations in the Plucker coordinates of $\mathbb G$ and covered by open subsets given by the non-vanishing of a Plucker coordinate, each embedded as a closed subscheme of the affine space $A^D$, $D=\dim(\mathbb G)$. However, the number $E$ of Plucker coordinates is so large that effective computations in this setting are practically impossible. In this paper, taking advantage of the symmetries of \hilb^n_p(t), we exhibit a new open cover, consisting of marked schemes over Borel-fixed ideals, whose number is significantly smaller than $E$. Exploiting the properties of marked schemes, we prove that these open subsets are defined by equations of degree $\leq d+2$ in their natural embedding in \Af^D. Furthermore we find new embeddings in affine spaces of far lower dimension than $D$, and characterize those that are still defined by equations of degree $\leq d+2$. The proofs are constructive and use a polynomial reduction process, similar to the one for Grobner bases, but are term order free. In this new setting, we can achieve explicit computations in many non-trivial cases.Comment: 17 pages. This version contains and extends the first part of version 2 (arXiv:0909.2184v2[math.AG]). A new extended version of the second part, with some new results, is posed at arxiv:1110.0698v3[math.AC]. The title is slightly changed. Final version accepted for publicatio