Groebner Basis Research Articles

Estimating the number of tetrahedra determined by volume, circumradius and four face areas using Groebner basis

Given any set of six positive parameters, the number of tetrahedra, all having these values as their volume, circumradius and four face areas, is studied. We identify all parameters that determine infinitely many tetrahedra. On the other hand, we classify parameters that determine finitely many tetrahedra and find only four different upper bounds, zero, six, eight, and nine, on the numbers of tetrahedra. In each case, the upper bound is sharp in the complex domain.In this paper, the upper bounds are obtained through checking the dimensions of various quotient algebras of ideals by counting monomials. This is done by computing Groebner bases with block orders. Partitioning the parameter space into several cases, we find either the dimension or an upper bound of it for the quotient algebra in each case. From that, various upper bounds on the number of tetrahedra are obtained. To show the upper bounds are sharp, we pick rational parameters and study the number of tetrahedra through Hermite's root counting method.

A degree-increasing [N to N+1] homotopy for Chebyshev and Fourier spectral methods

Hitherto, as a tool for tracing all branches of nonlinear differential equations, resolution-increasing homotopy methods have been applied only to finite difference discretizations. However, spectral Galerkin algorithms typically match the error of fourth order differences with one-half to one-fifth the number of degrees of freedom N in one dimension, and a factor of eight to a hundred and twenty-five in three dimensions. Let u → N be the vector of spectral coefficients and R → N the vector of N Galerkin constraints. A common two-part procedure is to first find all roots of R → N ( u → N ) = 0 → using resultants, Groebner basis methods or block matrix companion matrices. (These methods are slow and ill-conditioned, practical only for small N .) The second part is to then apply resolution-increasing continuation. Because the number of solutions is an exponential function of N , spectral methods are exponentially superior to finite differences in this context. Unfortunately, u → N is all too often outside the domain of convergence of Newton’s iteration when N is increased to ( N + 1 ) . We show that a good option is the artificial parameter homotopy H → ( u → ; τ ) ≡ R → N + 1 ( u → ) − ( 1 − τ ) R → N + 1 ( u → N ) , τ ∈ [ 0 , 1 ] . Marching in small steps in τ , we proceed smoothly from the N -term to the N + 1 -term approximations.

Solving positioning problems with minimal data

Global and local positioning systems (LPS) make use of nonlinear equations systems to calculate coordinates of unknown points. There exist several methods, such as Sturmfels' resultant, Groebner bases and least squares, for dealing with this kind of equations. We introduce two methods for solving this problem with the aid of symbolic techniques relying on closed-form solutions for the solution set of a system of linear equations. We suppose the receiver just detects or chooses minimal data, i.e., four satellites in global positioning systems (GPS) or three stations in LPS. Both methods proceed by parameterizing the line joining two solution points to later solve a nonlinear univariate equation, either quadratic or with degree smaller than 6. The first one uses the Generalized Cramer Identities, which is a different presentation of the generalized Moore---Penrose inverse, and ends with a degree 6 univariate equation for GPS and a degree 4 univariate equation for LPS. The second one solves the system by dealing with a more geometric way, ending with a quadratic equation. Our approach covers all possible cases with a finite number of solutions, while Bancroft's method cannot be applied when the four satellites, taking the clock bias as fourth coordinate, and the origin lay in the same hyperplane of $${\mathbb{R}}^{4}$$R4, and the method by Grafarend and Shan fails when the four satellites are in the same plane in $${\mathbb{R}}^{3}$$R3. The two proposed methods fail only when the pseudorange 4 point problem has infinite solutions.

On Computing Differential-Difference Groebner Bases with Two Term Order

On Computing Differential-Difference Groebner Bases with Two Term Order

A Groebner Bases Theory-Based Method for Selective Harmonic Elimination

An algebraic method is proposed for selective harmonic elimination PWM (SHEPWM). By computing its Groebner bases under the pure lexicographic monomial order, the nonlinear high-order SHE equations are converted to an equivalent triangular form, and then a recursive algorithm is used to solve the triangular equations one by one. Based on the proposed method, a user-friendly software package has been developed and some computation results are given. Unlike the commonly used numerical and intelligent methods, this method does not need to choose the initial values and can find all the solutions. Also, this method can give a definite answer to the question of whether the SHE equations have solutions or not, and the accuracy of the solved switching angles are much higher than that of the reference method. Compared with the existing algebraic methods, such as the resultant elimination method, the calculation efficiency is improved. Experimental verification is also shown in this paper.

Canonical Rings of $${\mathbb{Q}}$$ Q -Divisors on $${\mathbb{P}^1}$$ P 1

The canonical ring $S_D = \bigoplus_{d \geq 0} H^0(X, \lfloor dD \rfloor)$ of a divisor D on a curve X is a natural object of study; when D is a Q-divisor, it has connections to projective embeddings of stacky curves and rings of modular forms. We study the generators and relations of S_D for the simplest curve X = P^1. When D contains at most two points, we give a complete description of S_D; for general D, we give bounds on the generators and relations. We also show that the generators (for at most five points) and a Groebner basis of relations between them (for at most four points) depend only on the coefficients in the divisor D, not its points or the characteristic of the ground field; we conjecture that the minimal system of relations varies in a similar way. Although stated in terms of algebraic geometry, our results are proved by translating to the combinatorics of lattice points in simplices and cones.

Doing algebraic geometry with the RegularChains library

Traditionally, Groebner bases and cylindrical algebraic decomposition are the fundamental tools of computational algebraic geometry. Recent progress in the theory of regular chains has exhibited efficient algorithms for doing local analysis on algebraic varieties. In this note, we present the implementation of these new ideas within the module AlgebraicGeometryTools of the RegularChains library. The functionalities of this new module include the computation of the (non-trivial) limit points of the quasi-component of a regular chain. This type of calculation has several applications like computing the Zarisky closure of a constructible set as well as computing tangent cones of space curves, thus providing an alternative to the standard approaches based on Groebner bases and standard bases, respectively. From there, we have derived an algorithm which, under genericity assumptions, computes the intersection multiplicity of a zero-dimensional variety at any of its points. This algorithm relies only on the manipulations of regular chains.

Application of Dixon resultant to maximization of the likelihood function of Gaussian mixture distribution

In this presentation a new robust technique employing expectation maximization to separate outliers (corrupted data points) from inliers (true data points) iteratively, represented by different Gaussian distributions is introduced. Since in every iteration step, a new parameter estimation should be carried out, it is important to solve this parameter estimation as fast as possible. To do that, the problem of numerical global maximization of the likelihood function of the Gaussian mixture was transformed into the solution of a multivariate polynomial system. The symbolic solution of the resulting polynomial system consisting of four equations is quite challenging due to the high number of parameters. In order to solve it, a linear transformation was employed to reduce the number of the equations as well as the total degrees of the polynomials. This reduced system has been solved successfully via Dixon resultant accelerated by the Early Discovery of Factors method implemented in Fermat. The symbolic result was verified via numerical Groebner basis and the suggested robust technique was compared with other robust methods such as Danish and Random Sample Consensus methods based on the data set from a real laser scanning experiment.

Polynomial optimization with applications to stability analysis and control - Alternatives to sum of squares

In this paper, we explore the merits of various algorithms for solving polynomial optimization and optimization of polynomials, focusing on alternatives to sum of squares programming. While we refer to advantages and disadvantages of Quantifier Elimination, Reformulation Linear Techniques, Blossoming and Groebner basis methods, our main focus is on algorithms defined by Polya's theorem, Bernstein's theorem and Handelman's theorem. We first formulate polynomial optimization problems as verifying the feasibility of semi-algebraic sets. Then, we discuss how Polya's algorithm, Bernstein's algorithm and Handelman's algorithm reduce the intractable problem of feasibility of semi-algebraic sets to linear and/or semi-definite programming. We apply these algorithms to different problems in robust stability analysis and stability of nonlinear dynamical systems. As one contribution of this paper, we apply Polya's algorithm to the problem of $H_\infty$ control of systems with parametric uncertainty. Numerical examples are provided to compare the accuracy of these algorithms with other polynomial optimization algorithms in the literature.

Markov–chain Monte Carlo methods for the Box–Behnken designs and centrally symmetric configurations

We consider Markov chain Monte Carlo methods for calculating conditional p values of statistical models for count data arising in Box-Behnken designs. The statistical model we consider is a discrete version of the first-order model in the response surface methodology. For our models, the Markov basis, a key notion to construct a connected Markov chain on a given sample space, is characterized as generators of the toric ideals for the centrally symmetric configurations of root system D_n. We show the structure of the Groebner bases for these cases. A numerical example for an imaginary data set is given.

Local stability of cubic differential systems

Using Einstein’s notation [4]), the polynomial differential systems of finite dimension n and of degree at most k with coefficients in a field k of characteristic zero can be written as dx dt = a + ajα1x α1 + ajα1α2x 1x2 + ajα1···αrx α1 · · · xr (1) where j, α1, αr ∈ {1, . . . , n}, 1 ≤ r ≤ k, and for j = 1, . . . , n and for 2 ≤ r ≤ k, the tensor ajα1···αr (1 time contravariant and r times covariant) is symmetric with respect to the lower subscripts and we consider the action of the general linear group GL(n, k) on the phase space k. In this work, starting from a minimal system of generators of the ideal of algebraic invariants of polynomial differential systems (1) we will develop an algorithmic method to study the local stability of a given cubic differential system with the help of Grobner bases and using the GL(n, k)-invariants of this cubic system [2]. Our algorithmic method is motivated by the important role of the invariant theory in the survey of polynomial differential systems and the numerous works on Groebner basis one of the main practical tools for solving algebraic systems and computing algebraic varieties. Keywords Polynomial differential systems,linear transformations, cubic differential systems,singular points, invariant, covariant, Grobner bases.

Computing mobility condition using Groebner basis

A mechanism is a set of rigid links interconnected together with ideal joints and featuring at least one degree of freedom. An overconstrained mechanism is mobile provided the links' dimensions fulfill a certain relation named the “mobility condition”. The dimensions are not independent from each other and the goal is to obtain this mobility condition. Firstly, parameters are divided into two categories: dimensional parameters and positional parameters. Dimensional parameters represent links' sizes and positional parameters represent the relative positions between the links. The closure equation models the geometric problem by capturing the relationships between the two types of parameters. The principle of the paper is to generate the mobility condition by applying Groebner basis computation to the closure equation. Three methods are presented and investigated. The first one is called the univariate polynomial method (UNIPOL); the second method is called multi-order derivation method (MOD) and the third one is called the finitely separated configurations (FISECO) method. Practical implementation of these various techniques is explained by using a standard computer algebra system. The three methods are applied on a 2D overconstrained mechanism.

Computation of feedback control for time optimal state transfer using Groebner basis

Computation of time optimal feedback control law for a controllable linear time invariant system with bounded inputs is considered. Unlike a recent paper by the authors, the target final state is not limited to the origin of state-space, but is allowed to be in the set of constrained controllable states. Switching surfaces are formulated as semi-algebraic sets using Groebner basis based elimination theory. Using these semi-algebraic sets, a nested switching logic is synthesized to generate the time optimal feedback control. However, the optimal control law enforces an unavoidable limit cycle in the time-optimal trajectory for most non-origin target points. The time-period of this limit-cycle is dependent on the target position. This dependence is algebraically characterized and a method to compute the time-period of the limit-cycle is provided. As a natural extension, the set of constrained controllable states is also computed.

Determining “small parameters” for quasi-steady state

For a parameter-dependent system of ordinary differential equations we present a systematic approach to the determination of parameter values near which singular perturbation scenarios (in the sense of Tikhonov and Fenichel) arise. We call these special values Tikhonov–Fenichel parameter values. The principal application we intend is to equations that describe chemical reactions, in the context of quasi-steady state (or partial equilibrium) settings. Such equations have rational (or even polynomial) right-hand side. We determine the structure of the set of Tikhonov–Fenichel parameter values as a semi-algebraic set, and present an algorithmic approach to their explicit determination, using Groebner bases. Examples and applications (which include the irreversible and reversible Michaelis–Menten systems) illustrate that the approach is rather easy to implement.

Hochschild cohomology of cubic surfaces

We consider the polynomial algebra \({\mathbb{C}[{\bf z}] := \mathbb{C}[z_1, z_2, z_3]}\) and the polynomial \({f := z^{3}_{1} + z^3_2 + z^3_3 + 3qz_1z_2z_3}\), where \({q \in \mathbb{C}}\). Our aim is to compute the Hochschild homology and cohomology of the cubic surface \({\mathcal{X}_f := \{{\bf z} \in \mathbb{C}^3/f({\bf z}) = 0\}}\). For explicit computations, we shall make use of a method suggested by M. Kontsevich. Then, we shall develop it in order to determine the Hochschild homology and cohomology by means of multivariate division and Groebner bases. Some formal computations with Maple are also used.

Groebner basis based formal verification of large arithmetic circuits using Gaussian elimination and cone-based polynomial extraction

Verification of arithmetic circuits is essential as they form the main part of many practical designs such as signal processing and multimedia applications. In these applications, the size of the datapath could be very large so that contemporary verification methods would be almost incapable of verifying such circuits in reasonable time and memory usage. This paper addresses formal verification of large integer arithmetic circuits using symbolic computer algebra techniques. In order to efficiently verify gate level arithmetic circuits, we model the circuit and the specification with polynomial system and the verification problem is formulated as membership testing of the given specification polynomial in corresponding ideal of the circuit polynomials. The membership testing needs Groebner basis reduction. In order to overcome the intensive polynomial reduction needed in Groebner basis computation so that we can deal with verifying large arithmetic circuits, the fanout-free regions (cones) of the circuit are extracted and represented as corresponding polynomials automatically. For further improvement, we make use of Gaussian elimination concept to perform specification polynomial reduction w.r.t Groebner basis using a matrix representation of the problem. To evaluate the effectiveness of our verification technique, we have applied it to very large arithmetic circuits with different architectures. The experimental results show that the proposed verification technique is scalable enough so that large arithmetic circuits can efficiently be verified in reasonable run time and memory usage.

A matrix nullspace approach for solving equality-constrained multivariable polynomial least-squares problems

We present an elimination theory-based method for solving equality-constrained multivariable polynomial least-squares problems in system identification. While most algorithms in elimination theory rely upon Groebner bases and symbolic multivariable polynomial division algorithms, we present an algorithm which is based on computing the nullspace of a large sparse matrix and the zeros of a scalar, univariate polynomial.

On the computation of the Apéry set of numerical monoids and affine semigroups

A simple way of computing the Ap\'ery set of a numerical semigroup (or monoid) with respect to a generator, using Groebner bases, is presented, together with a generalization for affine semigroups. This computation allows us to calculate the type set and, henceforth, to check the Gorenstein condition which characterizes the symmetric numerical subgroups.

An Algebraic Approach to Fault Detection for Surge Avoidance in Turbo Compressor

This paper presents an innovative algebraic sensor fault detection approach for surge avoidance in turbo compressors (TC) in the natural gas compressor stations (GCS). The main objective is surge avoidance in the presence of sensor faults in TC. In this way, the robust parity space approach for fault detection is extended to highly nonlinear dynamic of TC based on Groebner basis and elimination technique. No work has been previously reported on the use of this technique for nonlinear dynamic systems with parametric uncertainties. This algebraic approach is simulated on the Moore–Greitzer control oriented model in the presence of parametric uncertainties, disturbances, and sensor faults. Simulation results are presented to demonstrate the effectiveness of the proposed fault detection approach.

Computing multiple periodic solutions of nonlinear vibration problems using the harmonic balance method and Groebner bases

This paper is devoted to the study of vibration of mechanical systems with geometric nonlinearities. The harmonic balance method is used to derive systems of polynomial equations whose solutions give the frequency component of the possible steady states. Groebner basis methods are used for computing all solutions of polynomial systems. This approach allows to reduce the complete system to an unique polynomial equation in one variable driving all solutions of the problem. In addition, in order to decrease the number of variables, we propose to first work on the undamped system, and recover solution of the damped system using a continuation on the damping parameter. The search for multiple solutions is illustrated on a simple system, where the influence of the retained number of harmonic is studied. Finally, the procedure is applied on a simple cyclic system and we give a representation of the multiple states versus frequency.