An extension of Buchberger's criteria for Groebner basis decision

Two fundamental questions in the theory of Groebner bases are decision ("Is a basis G of a polynomial ideal a Groebner basis?") and transformation ("If it is not, how do we transform it into a Groebner basis?") This paper considers the first question. It is well-known that G is a Groebner basis if and only if a certain set of polynomials (the S-polynomials) satisfy a certain property. In general there are m(m-1)/2 of these, where m is the number of polynomials in G, but criteria due to Buchberger and others often allow one to consider a smaller number. This paper presents two original results. The first is a new characterization theorem for Groebner bases that makes use of a new criterion that extends Buchberger's Criteria. The second is the identification of a class of polynomial systems G for which the new criterion has dramatic impact, reducing the worst-case scenario from m(m-1)/2 S-polynomials to m-1.Comment: 20 pages, 2 figure

Small Linearization: Memory Friendly Solving of Non-Linear Equations over Finite Fields

Solving non-linear and in particular Multivariate Quadratic equations over finite fields is an important cryptanalytic problem. Apart from needing exponential time in general, we also need very large amounts of memory, namely $\approx Nn^2$ for $n$ variables, solving degree $D$, and $N \approx n^D$. Exploiting systematic structures in the linearization matrix, we show how we can reduce this amount of memory by $n^2$ to $\approx N$. For practical problems, this is a significant improvement and allows to fit the overall algorithm in the RAM of \emph{one} machine, even for larger values of $n$. Hence we call our technique Small Linearization (sl). We achieve this by introducing a probabilistic version of the F$_5$ criterion. It allows us to replace (sparse) Gaussian Elimination by black box methods for solving the underlying linear algebra problem. Therefore, we achive a drastic reduction in the algorithm\u27s memory requirements. In addition, Small Linearization allows for far easier parallelization than algorithms using structured Gauss

Solving Systems of Multivariate Quadratic Equations over Finite Fields or: From Relinearization to MutantXL

In this article we investigate algorithms for solving non-linear multivariate equations over finite fields and the relation between them. For non binary fields usually computing the Gröbner basis of the corresponding ideal is the best choice in this context. One class of algorithms is based on Buchberger\u27s algorithm. Today\u27s best algorithms like F_4 and F_5 belong to this class. Another strategy to solve such systems is called eXtended Linearization (XL) from Eurocrypt 2000. In the past both strategies were treated as different ideas and there was a heated discussion which of them to prefer. Since Ars et al. proved in 2004 that XL is a redundant version of F_4, the latter seemed to be the winner. But that was not the end of the line as piece for piece the idea emerged that both classes are only different views on the same problem. We even think that they are just different time-memory optimizations. One indication to that can be found in the PhD of Albrecht, who introduced MatrixF_5, a F_5 version of XL. A second indication can be found in the PhD of Mohamed, who introduced a memory-friendly version of XL using Wiedemanns algorithm. We want to give further evidence by providing a theoretical analysis of MutantXL. We show that MutantXL solves at the same degree of regularity as its competitors F_4 and F_5 for most instances. Thereby we also confirm recent results of Albrecht, who showed that MutantXL is a redundant version of F_4, i.e. it never solves below the degree of regularity. We show that MutantXL has, compared to WiedemannXL, to pay its gain in efficiency with memory. To enhance the understanding of the whole XL-family of algorithms we give a full overview from Relinearization over XL to MutantXL and provide some additional theoretical insights

Unification in Commutative Theories, Hilbert's Basis Theorem, and Gröbner Bases

Unification in a commutative theory E may be reduced to solving linear equations in the corresponding semiring S(E) (Nutt (1988)). The unification type of E can thus be characterized by algebraic properties of S(E). The theory of abelian groups with n commuting homomorphisms corresponds to the semiring Z[X1,...,Xn]. Thus Hilbert’s Basis Theorem can be used to show that this theory is unitary. But this argument does not yield a unification algorithm. Linear equations in Z[X1,..,Xn] can be solved with the help of Gröbner Base methods, which thus provide the desired algorithm. The theory of abelian monoids with a homomorphism is of type zero (Baader (1988)). This can also be proved by using the fact that the corresponding semiring, namely N[X], is not noetherian. An other example of a semiring (even ring), which is not noetherian, is the ring Z, where X1, ..., Xn ( n > 1 ) are non-commuting indeterminates. This semiring corresponds to the theory of abelian groups with n non-commuting homomorphisms. Surprisingly, by construction of a Gröbner Base algorithm for right ideals in Z, it can be shown that this theory is unitary unifying

Averaging, reduction and reconstruction in the spatial three-body problem

El objetivo de esta tesis es el estudio de la dinámica del problema espacial de tres cuerpos. En particular, se establece la existencia de toros KAM asociados a diferentes tipos de movimientos. El problema espacial de tres cuerpos es un sistema hamiltoniano de nueve grados de libertad. La primera parte de la tesis consiste en aplicar técnicas de promedios y reducción con el fin de obtener un sistema reducido de un grado de libertad, es decir, aquel en el que todas las simetrías continuas han sido reducidas. El estudio, desarrollado a lo largo del presente documento, es válido en las regiones en las cuales el hamiltoniano del problema espacial de tres cuerpos puede ser expresado como suma de dos sistemas keplerianos más un pequeña perturbación. El proceso de reducción consta de las siguientes etapas: 1.- Reducción de la simetría traslacional. 2.- Reducción kepleriana, introducida en el proceso de normalización. 3.- Reducción de la simetría rotacional. 4.- Reducción de las simetría introducida al truncar el desarrollo del potencial. En primer lugar, reducimos la simetría traslacional, escribiendo el hamiltoniano en función de las coordenadas de Jacobi. A continuación, utilizamos las variables de Deprit para eliminar los nodos. Posteriormente, normalizamos con respecto de las anomalías medias en una región sin resonancias y truncamos los términos de mayor orden. El sistema obtenido es expresado en términos de los invariantes que definen el espacio reducido, el cual es una variedad simpléctica de dimensión ocho. En segundo lugar, se reduce la simetría rotacional que viene determinada por el hecho de que el módulo del momento angular total y su proyección en el eje vertical del sistema de referencia inercial son integrales del movimiento. Una vez calculados los invariantes asociados a las simetrías generadas por dichas integrales y el espacio reducido correspondiente, expresamos el hamiltoniano en término de estos invariantes. Ahora el espacio reducido tiene dimensión seis y es singular para algunos valores de los parámetros. En esta parte del estudio, la teoría de la reducción singular juega un papel clave. El último paso en el proceso de reducción es el de eliminar la simetría asociada al argumento del pericentro del cuerpo exterior. Dicha simetría aparece al truncar el hamiltoniano, puesto que este resulta ser independiente del argumento del pericentro. Una vez finalizado el proceso de reducción, obtenemos un espacio, que puede ser regular y difeomorfo a S2 o singular con a lo sumo tres puntos singulares, de dimensión dos parametrizado por medio de tres invariantes. En este espacio estudiamos los equilibrios relativos, su estabilidad y bifurcaciones. Partiendo del análisis de los equilibrios relativos en el espacio más reducido, llevamos a cabo la reconstrucción de toros KAM alrededor de cada equilibrio de tipo elíptico. Nuestro estudio consiste en una combinación de técnicas de regularizaci ón basadas en la construcción de espacios reducidos a diferentes niveles y la determinación explícita de coordenadas simplécticas. Todo esto nos permite calcular las torsiones para todas las posibles combinaciones de movimientos que las tres partículas puede seguir, incluyendo aquellos en los que los cuerpos interiores siguen trayectorias casi rectilíneas. Para probar la existencia de soluciones cuasi-periódicas utilizamos el teorema de Han, Li y Yi para sistemas hamiltonianos con alta degeneración y obtenemos toros KAM, de dimensión cinco, alrededor de equilibrios elípticos que representan diferentes tipos de movimientos. Centrándonos en los movimientos casi rectilíneos, encontramos soluciones cuasiperiódicas de los tres cuerpos tales que los dos cuerpos interiores describen órbitas cercanas a las de colisión. Los cuerpos interiores no colisionan, siguen órbitas acotadas con excentricidades próximas a uno. Estas soluciones están asociadas a puntos de equilibrio elípticos y o bien están en el plano invariable o son perpendiculares a él. Estas soluciones llenan toros invariantes de dimensión cinco.Programa Oficial de Doctorado en Métodos Matemáticos y sus Aplicaciones (RD 1393/2007)Metodo Matematikoetako eta beren Aplikazioetako Doktoretza Programa Ofiziala (ED 1393/2007

Berechnung und Anwendungen Approximativer Randbasen

This thesis addresses some of the algorithmic and numerical challenges associated with the computation of approximate border bases, a generalisation of border bases, in the context of the oil and gas industry. The concept of approximate border bases was introduced by D. Heldt, M. Kreuzer, S. Pokutta and H. Poulisse in "Approximate computation of zero-dimensional polynomial ideals" as an effective mean to derive physically relevant polynomial models from measured data. The main advantages of this approach compared to alternative techniques currently in use in the (hydrocarbon) industry are its power to derive polynomial models without additional a priori knowledge about the underlying physical system and its robustness with respect to noise in the measured input data. The so-called Approximate Vanishing Ideal (AVI) algorithm which can be used to compute approximate border bases and which was also introduced by D. Heldt et al. in the paper mentioned above served as a starting point for the research which is conducted in this thesis. A central aim of this work is to broaden the applicability of the AVI algorithm to additional areas in the oil and gas industry, like seismic imaging and the compact representation of unconventional geological structures. For this purpose several new algorithms are developed, among others the so-called Approximate Buchberger Möller (ABM) algorithm and the Extended-ABM algorithm. The numerical aspects and the runtime of the methods are analysed in detail - based on a solid foundation of the underlying mathematical and algorithmic concepts that are also provided in this thesis. It is shown that the worst case runtime of the ABM algorithm is cubic in the number of input points, which is a significant improvement over the biquadratic worst case runtime of the AVI algorithm. Furthermore, we show that the ABM algorithm allows us to exercise more direct control over the essential properties of the computed approximate border basis than the AVI algorithm. The improved runtime and the additional control turn out to be the key enablers for the new industrial applications that are proposed here. As a conclusion to the work on the computation of approximate border bases, a detailed comparison between the approach in this thesis and some other state of the art algorithms is given. Furthermore, this work also addresses one important shortcoming of approximate border bases, namely that central concepts from exact algebra such as syzygies could so far not be translated to the setting of approximate border bases. One way to mitigate this problem is to construct a "close by" exact border bases for a given approximate one. Here we present and discuss two new algorithmic approaches that allow us to compute such close by exact border bases. In the first one, we establish a link between this task, referred to as the rational recovery problem, and the problem of simultaneously quasi-diagonalising a set of complex matrices. As simultaneous quasi-diagonalisation is not a standard topic in numerical linear algebra there are hardly any off-the-shelf algorithms and implementations available that are both fast and numerically adequate for our purposes. To bridge this gap we introduce and study a new algorithm that is based on a variant of the classical Jacobi eigenvalue algorithm, which also works for non-symmetric matrices. As a second solution of the rational recovery problem, we motivate and discuss how to compute a close by exact border basis via the minimisation of a sum of squares expression, that is formed from the polynomials in the given approximate border basis. Finally, several applications of the newly developed algorithms are presented. Those include production modelling of oil and gas fields, reconstruction of the subsurface velocities for simple subsurface geometries, the compact representation of unconventional oil and gas bodies via algebraic surfaces and the stable numerical approximation of the roots of zero-dimensional polynomial ideals

Parallel Manipulators

In recent years, parallel kinematics mechanisms have attracted a lot of attention from the academic and industrial communities due to potential applications not only as robot manipulators but also as machine tools. Generally, the criteria used to compare the performance of traditional serial robots and parallel robots are the workspace, the ratio between the payload and the robot mass, accuracy, and dynamic behaviour. In addition to the reduced coupling effect between joints, parallel robots bring the benefits of much higher payload-robot mass ratios, superior accuracy and greater stiffness; qualities which lead to better dynamic performance. The main drawback with parallel robots is the relatively small workspace. A great deal of research on parallel robots has been carried out worldwide, and a large number of parallel mechanism systems have been built for various applications, such as remote handling, machine tools, medical robots, simulators, micro-robots, and humanoid robots. This book opens a window to exceptional research and development work on parallel mechanisms contributed by authors from around the world. Through this window the reader can get a good view of current parallel robot research and applications