Groebner Basis Research Articles

Mustafin Models of Projective Varieties and Vector Bundles

Abstract Mustafin varieties are well-studied degenerations of projective spaces induced by a choice of integral points in a Bruhat–Tits building. In recent work, Annette Werner and the author initiated the study of degenerations of plane curves obtained by Mustafin varieties by means of arithmetic geometry. Moreover, we applied these techniques to construct models of vector bundles on plane curves with strongly semistable reduction. In this work, we take a Groebner basis approach to the more general problem of studying degenerations of projective varieties. Our methods include determining the behaviour of Groebner bases under substitution over unique factorisation rings. Finally, we outline applications to the $p-$adic Simpson correspondence, when the respective projective variety is a curve.

English

In this paper, an original reduction algorithm for solving simultaneous multivariate polynomial equations is presented. The algorithm is exponential in complexity, but the well-known algorithms, such as the extended Euclidean algorithm and Buchberger's algorithm, are superexponential. The superexponential complexity of the well-known algorithms is due to their not being "minimal" in a certain sense. Buchberger's algorithm produces a Groebner basis. The proposed original reduction algorithm achieves the required task, via computation of determinants of parametric Sylvester matrices, and produces a Rabin basis, which is shown to be minimal, when two multivariate polynomials are reduced at a time. The minimality of Rabin basis allows us to prove exponential lower bounds for the space complexity of an algebraic proof of certification, for a specific computational problem in the computational complexity class PSPACE, showing that the complexity classes PSPACE and P cannot be the same. By the same reasoning, it follows that Co-NP is not the same as either of the complexity classes NP and P, and that the polynomial time hierarchy does not collapse. It is also shown that the class BPP of languages decidable by bounded error probabilistic algorithms with (probabilistic) polynomial time proofs for the membership of input words is not the same as any one of the complexity classes P, NP and Co-NP. It follows again, from the discussions, that the complexity classes NP and P are not the same, by relativization of BPP, with respect to P and NP.

Solution region synthesis methodology of spatial 1CS-4SS linkages for six given positions

This paper proposes a 1CS-4SS spatial linkage and presents the solution region synthesis methodology for six specified positions. Firstly, synthesis equations of the SS dyads and the CS dyad are proposed, the discrete solutions of the CS dyad are obtained by Bertini numerical method. The 1CS-4SS spatial linkage are obtained by selecting four SS dyads on the solution curves of the SS dyad and one CS dyad on the discrete solutions of the CS dyad. We display all solutions of linkages in a plane, which is called the solution region. Secondly, The Jacobian matrix is deduced to determinate the defective linkages. The Bertini numerical method and Groebner basis method are used for position analysis of the linkages, and the analysis results from the two methods are compared and mutually verified. The feasible solution region can be obtained after eliminating defective linkages and linkages that fail to satisfy the other requirements from the solution region plane. Finally, the validity of the formulas and applicability of the proposed approach are illustrated by an example.

Automatic selection of the Groebner Basis’ monomial order employed for the synthesis of the inverse kinematic model of non-redundant open-chain robotic systems

The methods most commonly used to synthesize the Inverse Kinematic Model (IKM) of open-chain robotic systems strongly depend on the robot’s geometry, which make them difficult to systematize. In a previous work we presented a systematic procedure that relies on Groebner Bases to synthesize the IKM of non-redundant open-chain robots. This study expands the developed procedure with a methodology for the automatic selection of the basis’ monomial order. The procedure’s inputs are the robot’s Denavit-Hartenberg parameters and the movement range of its actuators, while the output is the synthesized IKM, ready to be used in the robot’s control system or in a simulation of its behavior. This procedure can synthesize the IKM of a wide range of open-chain robotic systems, such as Cartesian robots, SCARA, non-redundant multi-legged robots, and all non-redundant manipulators that satisfy the in-line wrist condition. The procedure’s performance is assessed through two study cases of open-chain robots: a walking hexapod and a PUMA manipulator. The optimal monomial order is successfully identified for all cases. Also the output errors of the synthesized IKMs are negligible when evaluated in their corresponding workspaces, while their computation times are comparable to those required by the kinematic models calculated by traditional methods.

A New Approach to the Berlekamp-Massey-Sakata Algorithm: Improving Locator Decoding

We study the problem of the computation of Groebner basis for the ideal of linear recurring relations of a doubly periodic array. We find a set of indexes such that, along with some conditions, guarantees that the set of polynomials obtained at the last iteration in the Berlekamp-Massey-Sakata algorithm is exactly a Groebner basis for the mentioned ideal. Then, we apply these results to improve locator decoding in abelian codes.

The graph of a Weyl algebra endomorphism

Endomorphisms of Weyl algebras are studied using bimodules. Initially, for a Weyl algebra over a field of characteristic zero, Bernstein's inequality implies that holonomic bimodules finitely generated from the right or left form a monoidal category. The most important bimodule in this paper is the graph of an endomorphism. We prove that the graph of an endomorphism of a Weyl algebra over a field of characteristic zero is a simple bimodule. The simplicity of the tensor product of the dual graph and the graph is equivalent to the Dixmier conjecture. It is also shown how the graph construction leads to a non-commutative Groebner basis algorithm for detecting invertibility of an endomorphism for Weyl algebras and computing the inverse over arbitrary fields.

Algebraic immunity of vectorial boolean functions and boolean groebner bases

The basic concepts and results related to the Boolean Groebner bases and their application for computing the algebraic immunity of vectorial Boolean functions are considered. This parameter plays an important role for the security evaluation of block ciphers against algebraic attacks. Unlike the available works, the description is carried out at the elementary level using terms of Boolean functions theory. In addition, obtained proofs are shorter than the previous ones. This allows us to achieve significant progress in building the fundamentals of the theory (for the Boolean case) using only elementary methods.The paper can be useful for students and postgraduate students studying cryptology. It may also save time for professionals who want to get familiar with the mathematical techniques used in algebraic attacks on block ciphers.

Diagnosis in Tennis Serving Technique

Tennis is a sport with a very complex technique. Amateur tennis players have trainers and/or coaches, but are not usually accompanied by them to championships. Curiously, in this sport, the result of many matches can be changed by a small hint like ‘hit the ball a little higher when serving’. However, the biomechanical of a tennis stroke is only clear to an expert. We, therefore, developed a prototype of a rule-based expert system (RBES) aimed at an amateur competition player that is not accompanied by his/her coach to a championship and is not serving as usual (the RBES is so far restricted to serving). The player has to answer a set of questions about how he/she is serving that day and his/her usual serving technique and the RBES obtains a diagnosis using logic inference about the possible reasons (according of the logic rules that have been previously given to the RBES). A certain knowledge of the tennis terminology and technique is required from the player, but that is something known at this level. The underlying logic is Boolean and the inference engine is algebraic (it uses Groebner bases).

HELP: a sparse error locator polynomial for BCH codes

In 1990 Cooper suggested to use Groebner bases’ computations to decode cyclic codes and his idea gave rise to many research papers. In particular, as proved by Sala-Orsini, once defined the polynomial ring whose variables are the syndromes, the locations and the error values and considered the syndrome ideal, only one polynomial of a lexicographical Groebner basis for such ideal is necessary to decode (the general error locator polynomial, a.k.a. GELP). The decoding procedure only consists in evaluating this polynomial in the syndromes and computing its roots: the roots are indeed the error locations. A possible bottleneck in this procedure may be the evaluation part, since a priori the GELP may be dense. In this paper, focusing on binary cyclic codes with length $$n=2^m-1$$, correcting up to two errors, we give a Groebner-free, sparse analog of the GELP, the half error locator polynomial (HELP). In particular, we show that it is not necessary to compute the whole Groebner basis for getting such kind of locator polynomial and we construct the HELP, studying the quotient algebra of the polynomial ring modulo the syndrome ideal by a combinatorial point of view. The HELP turns out to be computable with quadratic complexity and it has linear growth in the length n of the code: $${\mathcal {O}}\left( \frac{n+1}{2}\right)$$.

Synthesis of the Inverse Kinematic Model of Non-Redundant Open-Chain Robotic Systems Using Groebner Basis Theory

One of the most important elements of a robot’s control system is its Inverse Kinematic Model (IKM), which calculates the position and velocity references required by the robot’s actuators to follow a trajectory. The methods that are commonly used to synthesize the IKM of open-chain robotic systems strongly depend on the geometry of the analyzed robot. Those methods are not systematic procedures that could be applied equally in all possible cases. This project presents the development of a systematic procedure to synthesize the IKM of non-redundant open-chain robotic systems using Groebner Basis theory, which does not depend on the geometry of the robot’s structure. The inputs to the developed procedure are the robot’s Denavit–Hartenberg parameters, while the output is the IKM, ready to be used in the robot’s control system or in a simulation of its behavior. The Groebner Basis calculation is done in a two-step process, first computing a basis with Faugère’s F4 algorithm and a grevlex monomial order, and later changing the basis with the FGLM algorithm to the desired lexicographic order. This procedure’s performance was proved calculating the IKM of a PUMA manipulator and a walking hexapod robot. The errors in the computed references of both IKMs were absolutely negligible in their corresponding workspaces, and their computation times were comparable to those required by the kinematic models calculated by traditional methods. The developed procedure can be applied to all Cartesian robotic systems, SCARA robots, all the non-redundant robotic manipulators that satisfy the in-line wrist condition, and any non-redundant open-chain robot whose IKM should only solve the positioning problem, such as multi-legged walking robots.

Simple Verification of Completeness of Two Addition Formulas on Twisted Edwards Curves

Daniel Bernstein and Tanja Lange proved that two given addition formulas on twisted Edwards elliptic curves ax^2 + y^2 = 1 + dxy are complete (i.e. the sum of any two points on a curve can be computed using one of these formulas). In this paper we give other simple verification of completeness of these formulas using for example Groebner bases and an ¨ algorithm implemented in Magma, which is based on the fact that completeness means that some systems of polynomial equations have no solutions. This method may be also applied to verify completeness of additions formulas on other models of elliptic curves.

Decomposition of polynomial sets into characteristic pairs

A characteristic pair is a pair (G,C) of polynomial sets in which G is a reduced lexicographic Groebner basis, C is the minimal triangular set contained in G, and C is normal. In this paper, we show that any finite polynomial set P can be decomposed algorithmically into finitely many characteristic pairs with associated zero relations, which provide representations for the zero set of P in terms of those of Groebner bases and those of triangular sets. The algorithm we propose for the decomposition makes use of the inherent connection between Ritt characteristic sets and lexicographic Groebner bases and is based essentially on the structural properties and the computation of lexicographic Groebner bases. Several nice properties about the decomposition and the resulting characteristic pairs, in particular relationships between the Groebner basis and the triangular set in each pair, are established. Examples are given to illustrate the algorithm and some of the properties.

Qualitative studies of some biochemical models

A computational approach to detecting Andronov–Hopf bifurcations in polynomial systems of ordinary differential equations depending on parameters is proposed. It relies on algorithms of computational commutative algebra based on the Groebner bases theory. The approach is applied to the investigation of two models related to the double phosphorylation of mitogen-activated protein kinases, a biochemical network that occurs in many cellular pathways. For the models, we analyze the roots of the characteristic polynomials of the Jacobians in a steady state and prove that Andronov–Hopf bifurcations are absent for biochemically relevant values of parameters. We also performed a search for algebraic invariant subspaces in the systems (which represent “weak” conservations laws) and find all subfamilies admitting linear invariant subspaces. The search is done using the Darboux method. That, is we look for Darboux polynomials and cofactors as polynomials with undetermined coefficients and then determine the coefficients using the algorithms of the elimination theory.

On Reidemeister torsion of flag manifolds of compact semisimple Lie groups

In this paper we calculate Reidemeister torsion of flag manifold $K/T$ of compact semi-simple Lie group $K = SU_{n+1}$ using Reidemeister torsion formula and Schubert calculus, where $T$ is maximal torus of $K$. We find that this number is 1. Also we explicitly calculate ring structure of integral cohomology algebra of flag manifold $K/T$ of compact semi-simple Lie group $K = SU_{n+1 }$ using root data, and Groebner basis techniques.

Exploration of Dual Curves Using a Dynamic Geometry and Computer Algebra System

This submission presents a particular example of the use of the free dynamic mathematics software GeoGebra to determine the dual curve to the given curve and to inspect its properties. The example is aimed at students of mathematics teaching. The combination of dynamic geometry tools with computer algebra functions, namely the functions for the computation of Groebner bases for polynomial ideals and a function for eliminating variables from the system of polynomial equations, based also on the method of Groebner bases, allows them to apply naturally the knowledge they acquire during their study of mathematics.

Groebner Basis and its Applications

This paper is a survey on Groebner basis and its applications on some areas of Science and Technology. Here we have presented some of the applications of concepts and techniques from Groebner basis to broader area of science and technology such as applications in steady state detection of chemical reaction network (CRN) by determining kinematics equations in the investigation and design of robots. Groebner basis applications could be found in vast area in circuits and systems. In pure mathematics, we can encounter many problems using Groebner basis to determine that a polynomial is invertible about an ideal, to determine radical membership, zero divisors, hence so forth. A short note is being presented on Groebner basis and its applications.

Multi-holographic metamaterials: The concept, artificially intelligent designs, and new applications in acoustics

Acoustic holographic metamaterials present a powerful new platform for manipulating acoustic wave fields in fluids and solids. Furthering their development, we introduce a new concept of multi-holographic metamaterials (MHM)—complex media storing multiple volumetric holograms. Such media lack any distributed electromechanical components, yet offer dynamic functionality using only a few electronic components. This novel engineering paradigm shifts device fabrication cost from hardware to computation, presenting the designer with an NP-hard problem of resolving the complex relationship between structure and physical properties. Focused on airborne ultrasonic metamaterials, we treat them as a free-form composite of a fluid and solid(s). This unlocks a huge number of structural degrees of freedom, compared to the standard treatment of metamaterials as periodic arrays of geometrically similar unit cells. While analytical methods for finding all optima of an analytical function are scarce and largely impractical (Groebner basis computation being a notable example), we turn to machine learning methodologies to design MHMs efficiently. We show that unsupervised machine learning, a basic form of Artificial Intelligence, can completely replace human input into the design of MHMs. Experimental demonstrations include airborne ultrasonic metamaterials for coherent beamforming, with applications ranging from directive speakers to contactless manipulation of objects.

Estimation of Source Wavelet From Seismic Traces Using Groebner Bases

An accurate and effective seismic wavelet estimation technique has extreme significance in the seismic data processing for analyzing the earth’s subsurface layer information. The seismic wavelet to be determined is modeled as a moving average (MA) process and assumed to be driven by a zero mean, non-Gaussian, statistically independent, and identically distributed (IID) process. In order to estimate the MA model parameter from the observed noisy seismic signal, we pose this as a blind system identification (BSI) problem. In the BSI, a set of multivariate polynomial equations is obtained by matching higher order cumulant of observed noisy data with a higher order moment of blind system’s impulse response. The Groebner bases that form the solution to this set of equations are obtained using the proposed algorithm. Numerical results demonstrate that the proposed method has a lower estimation error as compared to the previously reported methods.

Groebner bases algorithm for optimal PMU placement

In this paper a new method for solving optimal phasor measurement units (PMU) placement problem for full observability in power systems is proposed. A new algorithm is based on forming and solving Groebner bases set of equations for a defined optimization problem. The technique of Groebner bases is applicable wherever the problem may be presented by the set of polynomials and as such, it can be found in a variety of applied mathematics problems. Groebner bases of the set of polynomials represents the canonical form equivalent to the original set, similar to the way the row echelon form for linear systems represents the triangular form of the linear equations set. The canonical form of the Groebner bases polynomials, along with the utilization of recursive algorithms, allows the solution to be effortlessly obtained. Optimal phasor measurement units placement (OPP) problem can be presented in a linear or nonlinear polynomial form, both of which are used in this paper. The settings of OPP problem have been proposed in this paper with the inclusion of zero injection buses (ZIBs), in normal system states and considering contingency related to PMU outage. The problem formulation is defined according to the application of Groebner bases technique. The proposed model is capable of obtaining the full set of optimal solutions instead of only one or partial solutions. The efficiency of the proposed method is tested on IEEE 14-bus, IEEE 24-bus, IEEE 30-bus, IEEE 118-bus and Polish 2383-bus test system and the results are compared with those of some recently reported methods.

On the annihilator ideal of an inverse form: addendum

We improve results and proofs of our earlier paper and show that the ideal in question is a complete intersection. Let $${\mathbb {K}}$$ be a field and $$\mathrm {M}={\mathbb {K}}[x^{-1},z^{-1}]$$ denote the $${\mathbb {K}}[x,z]$$ submodule of Macaulay’s inverse system $${\mathbb {K}}[[x^{-1},z^{-1}]]$$ . We regard $$z\in {\mathbb {K}}[x,z]$$ and $$z^{-1}\in \mathrm {M}$$ as homogenising variables. An inverse form $$F\in \mathrm {M}$$ has a homogeneous annihilator ideal $${\mathcal {I}}_F$$ . In our earlier paper we inductively constructed an ordered pair ( $$f_1$$ , $$f_2$$ ) of forms in $${\mathbb {K}}[x,z]$$ which generate $${\mathcal {I}}_F$$ . We give a significantly shorter proof that accumulating all forms for F in our construction yields a minimal grlex Groebner basis $${\mathcal {F}}$$ for $${\mathcal {I}}_F$$ (without using the theory of S polynomials or distinguishing three types of inverse forms) and we simplify the reduction of $${\mathcal {F}}$$ . The associated Groebner basis algorithm terminates by construction and is quadratic. Finally we show that $$f_1,f_2$$ is a maximal $${\mathbb {K}}[x,z]$$ regular sequence for $${\mathcal {I}}_F$$ , so that $${\mathcal {I}}_F$$ is a complete intersection.