Rational Parametrization Research Articles

Toric difference variety

In this paper, the concept of toric difference varieties is defined and four equivalent descriptions for toric difference varieties are presented in terms of difference rational parametrization, difference coordinate rings, toric difference ideals, and group actions by difference tori. Connections between toric difference varieties and affine ℕ[x]-semimodules are established by proving the one-to-one correspondence between irreducible invariant difference subvarieties and faces of ℕ[x]-semimodules and the orbit-face correspondence. Finally, an algorithm is given to decide whether a binomial difference ideal represented by a ℤ[x]-lattice defines a toric difference variety.

The Arithmetic Geometry of Resonant Rossby Wave Triads

Linear wave solutions to the Charney--Hasegawa--Mima equation with periodic boundary conditions have two physical interpretations: Rossby (atmospheric) waves, and drift (plasma) waves in a tokamak. These waves display resonance in triads. In the case of infinite Rossby deformation radius, the set of resonant triads may be described as the set of integer solutions to a particular homogeneous Diophantine equation, or as the set of rational points on a projective surface $X$. The set of all resonant triads was found by Bustamante and Hayat (2013) via mapping to quadratic forms. Our work independently finds all resonant triads via a rational parametrization of $X$. We provide a fiberwise description of $X$ as a rational singular elliptic surface, yielding many new results about the set of wavevectors belonging to resonant triads. In particular, we show there is an infinite number of resonant triads (with relatively prime wavevectors) containing a wavevector $(a,b)$ with $a/b=r$, where $r$ is any given rationa...

Strong $\mu$-Bases for Rational Tensor Product Surfaces and Extraneous Factors Associated to Bad Base Points and Anomalies at Infinity

We investigate conditions under which the resultant of a $\mu$-basis for a rational tensor product surface is the implicit equation of the surface without any extraneous factors. In this case, we also derive a formula for the implicit degree of the rational surface based only on the bidegree of the rational parametrization and the bidegrees of the elements of the $\mu$-basis without any knowledge of the number or multiplicities of the base points, assuming only that all the base points are local complete intersections. We conclude that in this case the implicit degree of a rational surface of bidegree $(m,n)$ is at most $mn$, so the rational surface must have at least $mn$ base points counting multiplicity. When the resultant of a $\mu$-basis generates extraneous factors, we show how to predict and compute these extraneous factors from either the existence of bad base points or anomalies occurring in the parametrization at infinity. Examples are provided to flesh out the theory.

Detecting When an Implicit Equation or a Rational Parametrization Defines a Conical or Cylindrical Surface, or a Surface of Revolution.

Given an implicit polynomial equation or a rational parametrization, we develop algorithms to determine whether the set of real and complex points defined by the equation, i.e., the surface defined by the equation, in the sense of Algebraic Geometry, is a cylindrical surface, a conical surface, or a surface of revolution. The algorithms are directly applicable to, and formulated in terms of, the implicit equation or the rational parametrization. When the surface is cylindrical, we show how to compute the direction of its rulings; when the surface is conical, we show how to compute its vertex; and when the surface is a surface of revolution, we show how to compute its axis of rotation directly from the defining equations.

Micropolar shells with thickness inflation and independent micro‐rotations

Starting from constrained three‐dimensional Cosserat elasticity and in a total Lagrangian framework, we derive and discuss a hierarchy of static non‐linear micropolar shell models in which the thickness can stretch, and the independence of the micro‐rotation on the macro‐rotation is not limited to the drilling degree of freedom as in the classical six‐parameter theory. If the thickness stretch is considered, unconstrained quaternions allow a singularity‐free, non redundant and rational parametrization of the strain measures. Emphasis is placed in the discussion of the differences between curvatures, wryness and director deformation strains.

Frequency dispersion of small-amplitude capillary waves in viscous fluids.

This work presents a detailed study of the dispersion of capillary waves with small amplitude in viscous fluids using an analytically derived solution to the initial value problem of a small-amplitude capillary wave as well as direct numerical simulation. A rational parametrization for the dispersion of capillary waves in the underdamped regime is proposed, including predictions for the wave number of critical damping based on a harmonic-oscillator model. The scaling resulting from this parametrization leads to a self-similar solution of the frequency dispersion of capillary waves that covers the entire underdamped regime, which allows an accurate evaluation of the frequency at a given wave number, irrespective of the fluid properties. This similarity also reveals characteristic features of capillary waves, for instance that critical damping occurs when the characteristic time scales of dispersive and dissipative mechanisms are balanced. In addition, the presented results suggest that the widely adopted hydrodynamic theory for damped capillary waves does not accurately predict the dispersion when viscous damping is significant, and an alternative definition of the damping rate, which provides consistent accuracy in the underdamped regime, is presented.

Solving bivariate systems using Rational Univariate Representations

Given two coprime polynomials P and Q in Z[x,y] of degree bounded by d and bitsize bounded by τ, we address the problem of solving the system {P,Q}. We are interested in certified numerical approximations or, more precisely, isolating boxes of the solutions. We are also interested in computing, as intermediate symbolic objects, rational parameterizations of the solutions, and in particular Rational Univariate Representations (RURs), which can easily turn many queries on the system into queries on univariate polynomials. Such representations require the computation of a separating form for the system, that is a linear combination of the variables that takes different values when evaluated at the distinct solutions of the system.We present new algorithms for computing linear separating forms, RUR decompositions and isolating boxes of the solutions. We show that these three algorithms have worst-case bit complexity O˜B(d6+d5τ), where O˜ refers to the complexity where polylogarithmic factors are omitted and OB refers to the bit complexity. We also present probabilistic Las Vegas variants of our two first algorithms, which have expected bit complexity O˜B(d5+d4τ). A key ingredient of our proofs of complexity is an amortized analysis of the triangular decomposition algorithm via subresultants, which is of independent interest.

Permutation invariant properties of primitive cubic quadruples

Based on a specific quadratic Hopf map between the Euclidean spaces of dimension four and three that is associated with Euler’s complete rational parameterization of the four cubes equation, we study the permutation invariant properties of the primitive integer cubic quadruples that solve this equation. Fixing the coordinate with maximum height and taking it positive, our main result describes the six positive primitive triples that leave it invariant under the inverted cubic map to this Hopf map and permute the remaining integer coordinates. The obtained invariant primitive triples are ordered in the so-called integer triple ordering, so that the minimum triple with respect to this ordering determines each primitive cubic quadruple uniquely. Implications for the counting and enumeration of all primitive cubic quadruples are mentioned. A list of all primitive cubic quadruples with positive maximum height below 100 and their minimum invariant triples is given. The relationship with the famous Taxicab and Cabtaxi numbers is also explained.

Genus Two Curves with Many Elliptic Subcovers

We determine all genus 2 curves, defined over ℂ, which have simultaneously degree 2 and 3 elliptic subcovers. The locus of such curves has three irreducible 1-dimensional genus zero components in ℳ2. For each component, we find a rational parametrization and construct the equation of the corresponding genus 2 curve and its elliptic subcovers in terms of the parameterization. Such families of genus 2 curves are determined for the first time. Furthermore, we prove that there are only finitely many genus 2 curves (up to ℂ-isomorphism) defined over ℚ, which have degree 2 and 3 elliptic subcovers also defined over ℚ.

Influence of the number and location of design parameters in the aerodynamic shape optimization of a transonic aerofoil and a wing through evolutionary algorithms and support vector machines

ABSTRACTSurrogate-based optimization (SBO) has recently found widespread use in aerodynamic shape design owing to its promising potential to speed up the whole process by the use of a low-cost objective function evaluation, to reduce the required number of expensive computational fluid dynamics simulations. However, the application of these SBO methods for industrial configurations still faces several challenges. The most crucial challenge nowadays is the ‘curse of dimensionality’, the ability of surrogates to handle a high number of design parameters. This article presents an application study on how the number and location of design variables may affect the surrogate-based design process and aims to draw conclusions on their ability to provide optimal shapes in an efficient manner. To do so, an optimization framework based on the combined use of a surrogate modelling technique (support vector machines for regression), an evolutionary algorithm and a volumetric non-uniform rational B-splines parameterization are applied to the shape optimization of a two-dimensional aerofoil (RAE 2822) and a three-dimensional wing (DPW) in transonic flow conditions.

A solution method for autonomous first-order algebraic partial differential equations

In this paper we present a procedure for solving first-order autonomous algebraic partial differential equations in an arbitrary number of variables. The method uses rational parametrizations of algebraic (hyper)surfaces and generalizes a similar procedure for first-order autonomous ordinary differential equations. In particular we are interested in rational solutions and present certain classes of equations having rational solutions. However, the method can also be used for finding non-rational solutions.

Algebras of general type: Rational parametrization and normal forms

For every algebraically closed field k of characteristic different from 2, we prove the following: (1) Finite-dimensional (not necessarily associative) k-algebras of general type of a fixed dimension, considered up to isomorphism, are parametrized by the values of a tuple of algebraically independent (over k) rational functions of the structure constants. (2) There exists an “algebraic normal form” to which the set of structure constants of every such algebra can be uniquely transformed by means of passing to its new basis—namely, there are two finite systems of nonconstant polynomials on the space of structure constants, {fi}i∈I and {bj}j∈J, such that the ideal generated by the set {fi}i∈I is prime and, for every tuple c of structure constants satisfying the property bj(c) ≠ 0 for all j ∈ J, there exists a unique new basis of this algebra in which the tuple c′ of its structure constants satisfies the property fi(c′) = 0 for all i ∈ I.

Bulk irreducibles of the modular group

As the 3-string braid group B3 and the modular group Γ are both of wild representation type one cannot expect a full classification of all their finite dimensional simple representations. Still, one can aim to describe 'most' irreducible representations by constructing for each d-dimensional irreducible component X of the variety iss n(Γ) classifying the isomorphism classes of semi-simple n-dimensional representations of Γ an explicit minimal étale rational map 𝔸d → X having a Zariski dense image. Such rational dense parametrizations were obtained for all components when n < 12 in [5]. The aim of the present paper is to establish such parametrizations for all finite dimensions n.

Computing separating linear forms for bivariate systems

A fundamental problem in computational geometry is the computation of the topology of a real algebraic plane curve given by its implicit equation, that is, the computation of a set of polygonal lines that approximates the curve while preserving its topology. A critical step in many algorithms computing the topology of a plane curve is the computation of the set of singular and extreme points (wrt x) of this curve, which is equivalent to the computation of the solutions of bivariate systems defined by the curve and some of its partial derivatives. In this presentation, we address the problem of computing a linear separating form of a system of two bivariate polynomials with integer coefficients, that is a linear combination of the variables that takes different values when evaluated at the distinct solutions of the system. In other words, a separating linear form defines a shear of the coordinate system that sends the algebraic system in generic position, in the sense that no two distinct solutions are vertically aligned. The computation of such linear forms is at the core of most algorithms that solve algebraic systems by computing rational parameterizations of the solutions and this is the bottleneck of these algorithms in terms of worst-case bit complexity. We first present for this problem, an algorithm of worst-case bit complexity OB(d 8 +dτ) where d and τ denote respectively the maximum degree and bitsize of the input polynomials (and where O refers to the complexity where polylogarithmic factors are omitted and OB refers to the bit complexity). This complexity improves by a factor d the best known complexity bound for this problem (in OB(d 10 + dτ)). Then, we show how, using a similar approach, we can obtain an algorithm that simplifies the previous one while decreasing its complexity to OB(d 7 + dτ) bit operations in the worst-case. This new algorithm also yields a probabilistic Las-Vegas algorithm of expected bit complexity OB(d 5 + dτ).

Recognizing implicitly given rational canal surfaces

It is still a challenging task of today to recognize the type of a given algebraic surface which is described only by its implicit representation. In this paper we will investigate in more detail the case of canal surfaces that are often used in geometric modelling, Computer-Aided Design and technical practice (e.g. as blending surfaces smoothly joining two parts with circular ends). It is known that if the squared medial axis transform is a rational curve then so is also the corresponding surface. However, starting from a polynomial it is not known how to decide if the corresponding algebraic surface is a rational canal surface or not. Our goal is to formulate a simple and efficient algorithm whose input is a polynomial with the coefficients from some subfield of R and the output is the answer whether the surface is a rational canal surface. In the affirmative case we also compute a rational parameterization of the squared medial axis transform which can be then used for finding a rational parameterization of the corresponding implicitly given canal surface.

Complex μ-bases for real quadric surfaces

We provide a simple, efficient technique for computing μ-bases for quadric surfaces from their rational quadratic parametrizations. Our major innovation is to simplify the computations by using complex parameters, even though all the surfaces we treat have only real coefficients in both their implicit and parametric representations. In addition to the theory, we provide several examples to illustrate our method.

A new method to compute the singularities of offsets to rational plane curves

Given a planar curve defined by means of a real rational parametrization, we prove that the affine values of the parameter generating the real singularities of the offset are real roots of a univariate polynomial that can be derived from the parametrization of the original curve, without computing or making use of the implicit equation of the offset. By using this result, a finite set containing all the real singularities of the offset, and in particular all the real self-intersections of the offset, can be computed. We also report on experiments carried out in the computer algebra system Maple, showing the efficiency of the algorithm for moderate degrees.

Determination and (re)parametrization of rational developable surfaces

The developable surface is an important surface in computer aided design, geometric modeling and industrial manufactory. It is often given in the standard parametric form, but it can also be in the implicit form which is commonly used in algebraic geometry. Not all algebraic developable surfaces have rational parametrizations. In this paper, the authors focus on the rational developable surfaces. For a given algebraic surface, the authors first determine whether it is developable by geometric inspection, and then give a rational proper parametrization in the affirmative case. For a rational parametric surface, the authors also determine the developability and give a proper reparametrization for the developable surface.

Missing sets in rational parametrizations of surfaces of revolution

Parametric representations do not cover, in general, the whole geometric object that they parametrize. This can be a problem in practical applications. In this paper we analyze the question for surfaces of revolution generated by real rational profile curves, and we describe a simple small superset of the real zone of the surface not covered by the parametrization. This superset consists, in the worst case, of the union of a circle and the mirror curve of the profile curve.

On the rationality of the singularity locus of a Gough–Stewart platform — Biplanar case

We propose to study the singularity locus of a Gough–Stewart platform as a surface defined over the field of rational functions on the group of rotations. From the geometric properties of this surface we deduce, in the generic biplanar case, a rational parametrization and a family of parallel planes cutting the surface in the conics of a linear pencil, which are uniform for all generic orientations.