Trivariate Polynomial Research Articles

Improved Elekes-Szabó type estimates using proximity

We prove a new Elekes-Szabó type estimate on the size of the intersection of a Cartesian product A×B×C with an algebraic surface {f=0} over the reals. In particular, if A,B,C are sets of N real numbers and f is a trivariate polynomial, then either f has a special form that encodes additive group structure (for example, f(x,y,x)=x+y−z), or A×B×C∩{f=0} has cardinality O(N12/7). This is an improvement over the previous bound O(N11/6). We also prove an asymmetric version of our main result, which yields an Elekes-Ronyai type expanding polynomial estimate with exponent 3/2. This has applications to questions in combinatorial geometry related to the Erdős distinct distances problem.Like previous approaches to the problem, we rephrase the question as an L2 estimate, which can be analyzed by counting additive quadruples. The latter problem can be recast as an incidence problem involving points and curves in the plane. The new idea in our proof is that we use the order structure of the reals to restrict attention to a smaller collection of proximate additive quadruples.

Isolating all the real roots of a mixed trigonometric-polynomial

Mixed trigonometric-polynomials (MTPs) are functions of the form f(x,sin⁡x,cos⁡x) where f is a trivariate polynomial with rational coefficients, and the argument x ranges over the reals. In this paper, an algorithm “isolating” all the real roots of an MTP is provided and implemented. It automatically divides the real roots into two parts: one consists of finitely many roots in an interval [μ−,μ+] while the other consists of countably many roots in R﹨[μ−,μ+]. For the roots in [μ−,μ+], the algorithm returns isolating intervals and corresponding multiplicities while for those greater than μ+, it returns finitely many mutually disjoint small intervals Ii⊂[−π,π], integers ci>0 and multisets of root multiplicity {mj,i}j=1ci such that any root >μ+ is in the set (∪i∪k∈N(Ii+2kπ)) and any interval Ii+2kπ⊂(μ+,∞) contains exactly ci distinct roots with multiplicities m1,i,...,mci,i, respectively. The efficiency of the algorithm is shown by experiments. The method used to isolate the roots in [μ−,μ+] is applicable to any other bounded interval as well. The algorithm takes advantages of the weak Fourier sequence technique and deals with the intervals period-by-period without scaling the coordinate so to keep the length of the sequence short. The new approaches can easily be modified to decide whether there is any root, or whether there are infinitely many roots in unbounded intervals of the form (−∞,a) or (a,∞) with a∈Q.

Fast and accurate approximation of the angle-averaged redistribution function for polarized radiation

Context. Modeling spectral line profiles taking frequency redistribution effects into account is a notoriously challenging problem from the computational point of view, especially when polarization phenomena (atomic polarization and polarized radiation) are taken into account. Frequency redistribution effects are conveniently described through the redistribution function formalism, and the angle-averaged approximation is often introduced to simplify the problem. Even in this case, the evaluation of the emission coefficient for polarized radiation remains computationally costly, especially when magnetic fields are present or complex atomic models are considered. Aims. We aim to develop an efficient algorithm to numerically evaluate the angle-averaged redistribution function for polarized radiation. Methods. The proposed approach is based on a low-rank approximation via trivariate polynomials whose univariate components are represented in the Chebyshev basis. Results. The resulting algorithm is significantly faster than standard quadrature-based schemes for any target accuracy in the range [10−6, 10−2].

COORDINATISING PLANES OF PRIME POWER ORDER USING FINITE FIELDS

We revisit the coordinatisation method for projective planes by considering the consequences of using finite fields to coordinatise projective planes of prime power order. This leads to some general restrictions on the form of the resulting planar ternary ring (PTR) when viewed as a trivariate polynomial over the field. We also consider how the Lenz–Barlotti type of the plane being coordinatised impacts the form of the PTR polynomial, thereby deriving further restrictions.

New generalized expressions for forced convective heat transfer coefficients at building facades and roofs

Previous research indicated that the surface-averaged forced convective heat transfer coefficient (CHTC) at a windward building facade can vary substantially as a function of building width and height. However, existing CHTC expressions generally do not consider the building dimensions as parameters and are therefore strictly only applicable for the building geometry for which they were derived. Most CHTC expressions also categorize facades only as either windward or leeward. This indicates the need for new and more generally applicable CHTC expressions. This paper presents new generalized expressions for surface-averaged forced CHTC at building facades and roofs that contain the reference wind speed, the width and the height of the windward building facade as parameters. These expressions are derived from CFD simulations of wind flow and forced convective heat transfer for 81 different isolated buildings. The 3D Reynolds-averaged Navier-Stokes equations are solved with a combination of the high-Re number realizable k-ε model and the low-Re number Wolfshtein model. First, a validation study is performed with wind-tunnel measurements of surface temperature for a reduced-scale cubic model. Next, the actual simulations are performed on a high-resolution grid with a minimum near-wall cell size of 400 μm to resolve the entire boundary layer, including the viscous sublayer and the buffer layer, which dominate the convective surface resistance. The new CHTC expressions are analytical formulae (trivariate polynomials) that can easily be implemented in Building Energy Simulation (BES) and Building Envelope Heat-Air-Moisture (BE-HAM) transfer programs. The accuracy of the expressions is confirmed by in-sample and out-of-sample evaluations.

Trivariate polynomial approximation on Lissajous curves

We study Lissajous curves in the 3-cube, that generate algebraic cubature formulas on a special family of rank-1 Chebyshev lattices. These formulas are used to construct trivariate hyperinterpolation polynomials via a single 1-d Fast Chebyshev Transform (by the Chebfun package), and to compute discrete extremal sets of Fekete and Leja type for trivariate polynomial interpolation. Applications could arise in the framework of Lissajous sampling for MPI (Magnetic Particle Imaging).

On the Behavior of Minimal Free Resolutions of Trivariate Generic Monomial Ideals

We will explore some properties of minimal graded free resolutions of R/I, where R is a trivariate polynomial ring over a field and I is a monomial ideal. Our focus will be to consider a specific form of the resolutions when I is primary to the homogeneous maximal ideal. We will identify certain characteristics of the last matrix of these resolutions, and observe differences in the resolutions for generic ideals in comparison to nongeneric ideals. Finally, we learn how to identify whether I is generic by knowing the structure of the last matrix in the minimal free resolution of R/I.

ON THE UNIQUENESS FOR THE SYMMETRIC TENSOR RANK OF TRIVARIATE POLYNOMIALS; A LOCAL UNIQUENESS FOR MULTIVARIATE POLYNOMIALS

For all integers m ≥ 2 and d ≥ 3 and x > 0. Let νd : P m → P N , N := m+d m � − 1, be the Veronese embedding. We discuss the uniqueness (only for trivariate polynomials) and the local uniqueness of a decomposition of a polynomial into powers of linear forms in the following sense. Take P ∈ P N. Let S(m,x,d,P) be the set of all S ⊂ P m such that ♯(S) = x, P ∈ hνd(S)i (where h iis the linear span), and P /

THE DIRECT INVERSION OF HELIUM ATOM SCATTERING (HAS) DIFFRACTION SPECTRA

An algorithm that provides direct, efficient ND polynomial factorization is presented to solve the numerical issues that arise during the direct inversion of helium atom scattering (HAS) diffraction spectra. For an n-variate polynomial the algorithm directly deflates the polynomial to n-single variable equations by evaluating the ratio of pairs of polynomial coefficients. Error estimation of the coefficients of the 1D polynomials is then performed automatically using standard 1D search techniques. The effectiveness of the technique is demonstrated against bi- and trivariate polynomials and an approximate range of validity for error prone polynomials is demonstrated. To demonstrate the effectiveness of the technique, HAS diffraction spectra for the low coverage (2 × 1)-H/Pd(311) system have been analyzed using direct inversion and have revealed that H binds in a three-fold hollow site.

On Multivariate Chromatic Polynomials of Hypergraphs and Hyperedge Elimination

In this paper we introduce multivariate hyperedge elimination polynomials and multivariate chromatic polynomials for hypergraphs. The first set of polynomials is defined in terms of a deletion-contraction-extraction recurrence, previously investigated for graphs by Averbouch, Godlin, and Makowsky. The multivariate chromatic polynomial is an equivalent polynomial defined in terms of colorings, and generalizes the coboundary polynomial of Crapo, and the bivariate chromatic polynomial of Dohmen, Pönitz and Tittman. We prove that specializations of these new polynomials recover polynomials which enumerate hyperedge coverings, matchings, transversals, and section hypergraphs, all weighted according to certain statistics. We also prove that the polynomials can be defined in terms of Möbius inversion on the partition lattice of a hypergraph, and we compute these polynomials for various classes of hypergraphs. We also consider trivariate polynomials, which we call the hyperedge elimination polynomial and the trivariate chromatic polynomial.

Root lifting techniques and applications to list decoding

Motivatived by Guruswami and Rudra's construction of folded Reed-Solomon codes, we give algorithms to solve functional equations of the form Q(x, f(x), f(x)) = 0, where Q is a trivariate polynomial. We compare two approaches, one based on Newton's iteration and the second using relaxed series techniques.

A family of optimal sixteenth-order multipoint methods with a linear fraction plus a trivariate polynomial as the fourth-step weighting function

A new family of four-step optimal multipoint iterative methods of order sixteen for solving nonlinear equations are developed along with their convergence properties. Numerical experiments with comparison to some existing methods are demonstrated to support the underlying theory.

Generalized Duffy transformation for integrating vertex singularities

For an integrand with a 1/r vertex singularity, the Duffy transformation from a triangle (pyramid) to a square (cube) provides an accurate and efficient technique to evaluate the integral. In this paper, we generalize the Duffy transformation to power singularities of the form p(x)/r α , where p is a trivariate polynomial and α > 0 is the strength of the singularity. We use the map (u, v, w) → (x, y, z) : x = u β , y = x v, z = x w, and judiciously choose β to accurately estimate the integral. For α = 1, the Duffy transformation (β = 1) is optimal, whereas if α ≠ 1, we show that there are other values of β that prove to be substantially better. Numerical tests in two and three dimensions are presented that reveal the improved accuracy of the new transformation. Higher-order partition of unity finite element solutions for the Laplace equation with a derivative singularity at a re-entrant corner are presented to demonstrate the benefits of using the generalized Duffy transformation.

Soft-decision list decoding of hermitian codes

This paper proposes the first complete soft-decision list decoding algorithm for Hermitian codes based on the Koetter-Vardy's Reed-Solomon code decoding algorithm. For Hermitian codes, interpolation processes trivariate polynomials which are defined over the pole basis of a Hermitian curve. In this paper, the interpolated zero condition of a trivariate polynomial with respect to a multiplicity matrix M is redefined followed by a proof of the validity of the soft-decision scheme. This paper also introduces a new stopping criterion for the algorithm that tranforms the reliability matrix Pi to the multiplicity matrix M. Geometric characterisation of the trivariate monomial decoding region is investigated, resulting in an asymptotic optimal performance bound for the soft-decision decoder. By defining the weighted degree upper bound of the interpolated polynomial, two complexity reducing modifications are introduced for the soft-decision scheme: elimination of unnecessary interpolated polynomials and pre-calculation of the coefficients that relate the pole basis monomials to the zero basis functions of a Hermitian curve. Our simulation results and analyses show that soft-decision list decoding of Hermitian code can outperform Koetter-Vardy decoding of Reed-Solomon code which is defined in a larger finite field, but with less decoding complexity.

Small Secret Key Attack on a Takagi's Variant of RSA

For a variant of RSA with modulus N = prq and ed ≡ 1 (mod(p-1)(q-1)), we show that d is to be recovered if d < N(2-√2)/(r+1). (Note that φ(N) ≠ (p-1)(q-1).) Boneh-Durfee's result for the standard RSA is obtained as a special case for r=1. Technically, we develop a method for finding a small root of a trivariate polynomial equation f(x, y, z)=x(y-1)(z-1)+1 ≡ 0 (mod e) under the condition that yrz=N. Our result cannot be obtained from the generic method of Jochemsz-May.

Soft decoding of algebraic–geometric codes using Koetter-Vardy algorithm

The first soft-decision decoding algorithm for algebraic–geometric codes is presented. This algorithm is developed based on the Koetter-Vardy algorithm, which was proposed for the decoding of Reed-Solomon codes. The interpolation process is modified by redefining the zero condition of a trivariate polynomial and introducing complexity reducing methods. Simulation results show that 0.7 and 1.7 dB performance improvements over the hard-decision decoding bound can be achieved in AWGN and quasi-static Rayleigh fading channels, respectively.

Parallel methods for absolute irreducibility testing

A heuristic algorithm for testing absolute irreducibility of multivariate polynomials over arbitrary fields using Newton polytopes was proposed in Gao and Lauder (Discrete Comput. Geom. 26:89–104, [2001]). A preliminary implementation by Gao and Lauder (2003) established a wide range of families of low degree and sparse polynomials for which the algorithm works efficiently and with a high success rate. In this paper, we develop a BSP variant of the absolute irreducibility testing via polytopes, with the aim of producing a more memory and run-time efficient method that can provide wider ranges of applicability, specifically in terms of the degrees of the input polynomials. In the bivariate case, we describe a balanced load scheme and a corresponding data distribution leading to a parallel algorithm whose efficiency can be established under reasonably realistic conditions. This is later incorporated in a doubly parallel algorithm in the multivariate case that achieves similar scalable performance. Both parallel models are analyzed for efficiency, and the theoretical analysis is compared to the performance of our experiments. In the empirical results we report, we achieve absolute irreducibility testing for bivariate and trivariate polynomials of degrees up to 30,000, and for low degree multivariate polynomials with more than 3,000 variables. To the best of our knowledge, this sets a world record in establishing absolute irreducibility of multivariate polynomials.

Hardware‐Accelerated, High‐Quality Rendering Based on Trivariate Splines Approximating Volume Data

AbstractWe develop an approach for hardware‐accelerated, high‐quality rendering of volume data using trivariate splines. The proposed quasi‐interpolating schemes are realtime reconstructions. The low total degrees provide several advantages for our GPU implementation. In particular, intersecting rays with spline isosurfaces for direct Phong illumination is performed by simple root finding algorithms (analytic and iterative), while the necessary normals result from blossoming. Since visualizations are on a fragment base, our renderer for isosurfaces includes an automatic level of detail. While we use well‐known spatial data structures in the CPU part of the algorithm for hierarchical view frustum culling and memory reduction, our GPU implementations have to take the highly complex structure of the splines into account. These include an appropriate organization of the data streams, i.e. we develop an advanced encoding scheme for the spline coefficients, as well as an implicit scheme for bounding geometry retrieval. In addition, we propose an elaborated clipping procedure to be performed in the fragment shader. These features essentially reduce bus traffic, memory consumption, and data access on the GPU leading to interactive frame rates for renderings of high visual quality. Compared with pure CPU implementations and existing GPU implementations for trivariate polynomials frame rates increase by factors between 10 and 100.

Boolean Set Operations with Cubic Algebraic Patches

In this paper we discuss a symbolic-numeric algorithm for Boolean operations, closed in the algebra of curved polyhedra whose boundary is triangulated with algebraic patches (A-patches). This approach uses a linear polyhedron as a first approximation of both the arguments and the result. On each triangle of a boundary representation of such linear approximation, a piecewise cubic algebraic interpolant is built, using a C(1)-continuous prism algebraic patch (prism A-patch) that interpolates the three triangle vertices, with given normal vectors. The boundary representation only stores the vertices of the initial triangulation and their external vertex normals. In order to represent also flat and/or sharp local features, the corresponding normal-per-face and/or normal-per-edge may be also given, respectively. The topology is described by storing, for each curved triangle, the two triples of pointers to incident vertices and to adjacent triangles. For each triangle, a scaffolding prism is built, produced by its extreme vertices and normals, which provides a containment volume for the curved interpolating A-patch. When looking for the result of a regularized Boolean operation, the 0-set of a tri-variate polynomial within each such prism is generated, and intersected with the analogous 0-sets of the other curved polyhedron, when two prisms have non-empty intersection. The intersection curves of the boundaries are traced and used to decompose each boundary into the 3 standard classes of subpatches, denoted in, out and on. While tracing the intersection curves, the locally refined triangulation of intersecting patches is produced, and added to the boundary representation.

Numerical integration of some functions over an arbitrary linear tetrahedra in Euclidean three-dimensional space

In this paper it is proposed to compute the volume integral of certain functions whose antiderivates with respect to one of the variates (say either x or y or z) is available. Then by use of the well known Gauss Divergence theorem, it can be shown that the volume integral of such a function is expressible as sum of four integrals over the unit triangle. The present method can also evaluate the triple integrals of trivariate polynomials over an arbitrary tetrahedron as a special case. It is also demonstrated that certain integrals which are nonpolynomial functions of trivariates x , y , z can be computed by the proposed method. We have applied Gauss Legendre Quadrature rules which were recently derived by Rathod et al. [H.T. Rathod, K.V. Nagaraja, B. Venkatesudu, N.L. Ramesh, Gauss Legendre Quadrature over a Triangle, J. Indian Inst. Sci. 84 (2004) 183–188] to evaluate the typical integrals governed by the proposed method.