SOME APPLICATIONS OF GRÖBNER BASES (original) (raw)
Related papers
Gröbner Bases: An Introduction and Some Applications
International Virtual FDP on “Frontiers of Mathematics”, 2020
* Introduction ? Varieties ? Ideals ? Linear Case ? Polynomials of One Variable * Gröbner Bases ? Term Orders ? S-Polynomials ? Buchberger’s Algorithm ? Sample Computations * Some Application of Gröbner Bases ? The 3-Color Problem ? Automatic Geometric Theorem Proving ? Other Applications * References
New developments in the theory of Gröbner bases and applications to formal verification
Journal of Pure and Applied Algebra, 2009
We present foundational work on standard bases over rings and on Boolean Gröbner bases in the framework of Boolean functions. The research was motivated by our collaboration with electrical engineers and computer scientists on problems arising from formal verification of digital circuits. In fact, algebraic modelling of formal verification problems is developed on the word-level as well as on the bit-level. The word-level model leads to Gröbner basis in the polynomial ring over Z/2 n while the bit-level model leads to Boolean Gröbner bases. In addition to the theoretical foundations of both approaches, the algorithms have been implemented. Using these implementations we show that special data structures and the exploitation of symmetries make Gröbner bases competitive to state-of-the-art tools from formal verification but having the advantage of being systematic and more flexible.
Algorithmic Algebraic Combinatorics and Gröbner Bases
Algorithmic Algebraic Combinatorics and Gröbner Bases, 2009
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Towards the Automated Synthesis of a Grobner Bases Algorithm
Revista de la Real Academia de Ciencias …, 2004
We discuss the question of whether the central result of algorithmic Gröbner bases theory, namely the notion of S-polynomials together with the algorithm for constructing Gröbner bases using S-polynomials, can be obtained by "artificial intelligence", i.e. a systematic (algorithmic) algorithm synthesis method. We present the "lazy thinking" method for theorem and algorithm invention and apply it to the "critical pair / completion" algorithm scheme. We present a road map that demonstrates that, with this approach, the automated synthesis of the author's Gröbner bases algorithm is possible. Still, significant technical work will be necessary to improve the current theorem provers, in particular the ones in the Theorema system, so that the road map can be transformed into a completely computerized process. Hacia la síntesis automática de un algoritmo de bases de Gröbner Resumen. Se aborda la cuestión de si el resultado central de la teoría algorítmica de bases de Gröbner, es decir, la noción de S-polinomio, junto con el algoritmo de construcción de bases de Gröbner basado en S-polinomios, puede obtenerse mediante la "inteligencia artificial", es decir, por un método sistemático de síntesis algorítmica. En concreto, se presenta el método "lazy thinking" para la invención de teoremas y algoritmos, que se aplica al esquema algorítmico de "par crítico/completitud". Se presenta una "hoja de ruta" que demuestra que este enfoque permite la síntesis automática del algoritmo de bases de Gröbner del autor. No obstante, será necesario mejorar los actuales demostradores de teoremas y, sobre todo, los del sistema "Theorema", para que esa "hoja de ruta" se pueda transformar en un proceso completamente computerizado, lo que aún supondrá un trabajo técnico importante.
On the robust hardness of Gröbner basis computation
Journal of Pure and Applied Algebra
The computation of Gröbner bases is an established hard problem. By contrast with many other problems, however, there has been little investigation of whether this hardness is robust. In this paper, we frame and present results on the problem of approximate computation of Gröbner bases. We show that it is NP-hard to construct a Gröbner basis of the ideal generated by a set of polynomials, even when the algorithm is allowed to discard a 1 − ǫ fraction of the generators, and likewise when the algorithm is allowed to discard variables (and the generators containing them). Our results shows that computation of Gröbner bases is robustly hard even for simple polynomial systems (e.g. maximum degree 2, with at most 3 variables per generator). We conclude by greatly strengthening results for the Strong c-Partial Gröbner problem posed by De Loera et al. [10]. Our proofs also establish interesting connections between the robust hardness of Gröbner bases and that of SAT variants and graph-coloring.
Applicable Algebra in Engineering, Communication and Computing, 2006
In the first sections we extend and generalize Gröbner basis theory to submodules of free right modules over monoid rings. Over free monoids, we adapt the known theory for right ideals and prove versions of Macaulay's basis theorem, the Buchberger criterion, and the Buchberger algorithm. Over monoids presented by a finitely generated convergent string rewriting system we generalize Madlener's Gröbner basis theory based on prefix reduction from right ideals to right modules. After showing how these Gröbner basis theories relate to classical group-theoretic problems, we use them as a basis for a new class of cryptosystems that are generalizations of the cryptosystems described in [2] and [8]. Well known cryptosystems such as RSA, El-Gamal, Polly Cracker, Polly Two and a braid group cryptosystem are shown to be special cases. We also discuss issues related to the security of these Gröbner basis cryptosystems.
Gröbner bases for public key cryptography
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation - ISSAC '08, 2008
Up to now, any attempt to use Gröbner bases in the design of public key cryptosystems has failed, as anticipated by a classical paper of B. Barkee et al.; we show why, and show that the only residual hope is to use binomial ideals, i.e. lattices. We propose two lattice-based cryptosystems that will show the usefulness of multivariate polynomial algebra and Gröbner bases in the construction of public key cryptosystems. The first one tries to revive two cryptosystems Polly Cracker and GGH, that have been considered broken, through a hybrid; the second one improves a cryptosystem (NTRU) that only has heuristic and challenged evidence of security, providing evidence that the extension cannot be broken with some of the standard lattice tools that can be used to break some reduced form of NTRU. Because of the bounds on length, we only sketch the construction of these two cryptosystems, and leave many details of the construction of private and public keys, of the proofs and of the security considerations to forthcoming technical papers.
Extended Hardness Results for Approximate Gröbner Basis Computation
ArXiv, 2016
Two models were recently proposed to explore the robust hardness of Gr\"obner basis computation. Given a polynomial system, both models allow an algorithm to selectively ignore some of the polynomials: the algorithm is only responsible for returning a Gr\"obner basis for the ideal generated by the remaining polynomials. For the qqq-Fractional Gr\"obner Basis Problem the algorithm is allowed to ignore a constant (1−q)(1-q)(1−q)-fraction of the polynomials (subject to one natural structural constraint). Here we prove a new strongest-parameter result: even if the algorithm is allowed to choose a (3/10−epsilon)(3/10-\epsilon)(3/10−epsilon)-fraction of the polynomials to ignore, and need only compute a Gr\"obner basis with respect to some lexicographic order for the remaining polynomials, this cannot be accomplished in polynomial time (unless P=NPP=NPP=NP). This statement holds even if every polynomial has maximum degree 3. Next, we prove the first robust hardness result for polynomial systems of maximum de...
Geometric Characterization of Data Sets with Unique Reduced Gröbner Bases
Bulletin of Mathematical Biology
Applications of Gröbner bases, such as reverse engineering of gene regulatory networks and combinatorial encoding of receptive fields, consider data sets whose ideals of points have unique reduced Gröbner bases. The significance is that uniqueness provides a canonical representation of the input data. In this work, we identify geometric properties of input data that result in a unique reduced Gröbner basis. We show that if the data form a staircase or a so-called linear shift of a staircase, the ideal of the points has a unique reduced Gröbner basis. These results serve to minimize computational effort in using Gröbner bases and are a stepping stone for developing algorithms to generate such data sets.
A Note on GR Obner Bases and Graph Colorings
In this paper, we correct a minor misstatement in [4], where J.A. De Loera demonstrates an explicit universal Gröbner basis of the radical ideal of a variety related to chromatic numbers. We show that this result does not hold when the base field is finite, and we correct it for this case.