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AIRO INTERNATIONAL JOURNAL VOLUME 6 ISSN 23203714
An Introduction to Grobner Bases
Submitted by Rajwinder Kaur :
Research Scholar Mathematics
Abstract
One of the purposes of research paper is to show that, in many cases Gröbner bases can be used
to find solutions for formal verification problems. In this way, this forms a good complement to
existing techniques, like simulators and SAT-solver, which are suited for identification of
counter examples (falsification).
A significant advantage is, that Gröbner bases provide a mathematically proven systematic and
very flexible tool while many engineering solutions inside commercial verification tools rely on
ad hoc heuristics for special cases. However, the success of Gröbner basis methods, reported in
this paper, could not be achieved with existing generic Gröbner basis algorithms and
implementations.
Introduction
A new efficient algorithm for computing Gröbner bases which is to avoid as much intermediate
computation as possible, the algorithm computes successive truncated Gröbner bases and it replaces
the classical polynomial reduction found in the Buchberger algorithm by the simultaneous reduction of
several polynomials. This powerful reduction mechanism is achieved by means of a symbolic
precomputation and by extensive use of sparse linear algebra methods. Current techniques in linear
algebra used in Computer Algebra are reviewed together with other methods coming from the
numerical field. Some previously untractable problems (Cyclic 9) are presented as well as an empirical
comparison of a first implementation of this algorithm with other well known programs. This
comparison pays careful attention to methodology issues. All the benchmarks and CPU times used in
this paper are frequently updated and available on a Web page. Even though the new algorithm does
not improve the worst case complexity it is several times faster than previous implementations both for
integers and modulo p computations. a new efficient algorithm for computing Gröbner bases.
AIRO INTERNATIONAL JOURNAL VOLUME 6 ISSN 23203714
Review of literature
All operations related to Gröbner bases require the choice of a total order on the monomials,
with
the
following
properties
of
compatibility
with
multiplication.
For
all
monomials M, N, P,
1.
2.
.
A total order satisfying these condition is sometimes called an admissible ordering.
These conditions imply Noetherianity, which means that every strictly decreasing sequence
of monomials is finite.
Although Gröbner basis theory does not depend on a particular choice of an admissible
monomial ordering, three monomial orderings are specially important for the applications:

Lexicographical ordering, commonly called lex or plex (for pure lexical ordering).

Total degree reverse lexicographical ordering, commonly called degrevlex.

Elimination ordering, lexdeg.
Gröbner basis theory was initially introduced for the lexicographical ordering. It was soon
realised that the Gröbner basis for degrevlex is almost always much easier to compute, and
that it is almost always easier to compute a lex Gröbner basis by first computing the
degrevlex basis and then using a "change of ordering algorithm". When elimination is
needed, degrevlex is not convenient; both lex and lexdeg may be used but, again, many
computations are relatively easy with lexdeg and almost impossible with lex.
Once a monomial ordering is fixed, the terms of a polynomial (product of a monomial with
its nonzero coefficient) are naturally ordered by decreasing monomials (for this order). This
makes the representation of a polynomial as an ordered list of pairs coefficient–exponent
vector a canonical representation of the polynomials. The first (greatest) term of a
polynomial p for this ordering and the corresponding monomial and coefficient are
AIRO INTERNATIONAL JOURNAL VOLUME 6 ISSN 23203714
respectively called the leading term, leading monomial and leading coefficient and denoted,
in this article, lt(p), lm(p) and lc(p).
Reduction
The concept of reduction, also called multivariate division or normal form computation,
is central to Gröbner basis theory. It is a multivariate generalization of the Euclidean division
of univariate polynomials.
In this section we suppose a fixed monomial ordering, which will not be defined explicitly.
Given two polynomials f and g, one says that f is reducible by g if some monomial m in f is a
multiple of the leading monomial lm(g) of g. If m happens to be the leading monomial
of f then one says that f is lead-reducibleby g. If c is the coefficient of m in f and m = q lm(g),
the one-step reduction of f by g is the operation that associates to f the polynomial
The main properties of this operation are that the resulting polynomial does not contain
the monomial m and that the monomials greater than m (for the monomial ordering)
remain unchanged. This operation is not, in general, uniquely defined; if several
monomials in f are multiples of lm(g) one may choose arbitrarily the one that is reduced.
In practice, it is better to choose the greatest one for the monomial ordering, because
otherwise subsequent reductions could reintroduce the monomial that has just been
removed.
Given a finite set G of polynomials, one says that f is reducible or lead-reducible by G if
it is reducible or lead-reducible, respectively, by an element of G. If it is the case, then
one defines . The (complete) reduction of f by G consists in applying iteratively this
operator until getting a polynomial , which is irreducible by G. It is called a normal
form of f by G. In general this form is not uniquely defined (this is not a canonical form);
this non-uniqueness is the starting point of Gröbner basis theory.
For Gröbner basis computations, except at the end, it is not necessary to do a complete
reduction: a lead-reduction is sufficient, which saves a large amount of computation.
AIRO INTERNATIONAL JOURNAL VOLUME 6 ISSN 23203714
Research Methodology
The theory of Gröbner bases has been extended by many authors in various directions. It has
been generalized to other structures such as polynomials over principal ideal rings or polynomial
rings, and also some classes of non-commutative rings and algebras, like Ore algebras.
Polynomial Algebra
Certain sets of polynomials have special algebraic structure, they may be rings, fields or ideals. Algebraic
properties associated to these structures play a very im-portant role in solving computational tasks
involving polynomials. In this chapter, we will discuss various aspects of polynomials which will play a
fundamental role in our later discussion. We will define ideals over polynomial rings and will give brief
summary on Groebner bases for ideals and modules. Most of the material given in this section is taken
from [1, 18].
Monomials
A monomial, in n indeterminates x1, . . . , xn, is a product of the form xu11 xu22 · · · xunn , where the ui are
non-negative integers, and u = (u1, . . . , un). The total degree of this monomial is the sum |u| = u1 + · · · +
un.
2.1.1 Definition [Polynomial] A polynomial f in x1, x2, . . . , xn with the coeffi-cients in K (where K is any
field) is a finite linear combination of the monomials, written as
f = cuxu, cu ∈ K,
(2.1)
∑
u
cu is called the coefficient of the monomial xu = xu11 · · · xunn . If cu , 0 then we call cuxu a term of f . The set
of all polynomials in x1, x2, . . . , xn with coefficients in
AIRO INTERNATIONAL JOURNAL VOLUME 6 ISSN 23203714
is denoted by K[x] = K[x1, . . . , xn]. These polynomials in n variables, over a field K, together with
operations of addition and multiplication, satisfy all axioms of ring, and so form commutative
polynomial ring.
Objective
This research work is elaborating the methodology of Grobner and its application in various
mathematical aspects.
The main objective is to determine the more and more implementation of the theory of Grobner in the
world of Mathematics
Through this work we are to know the following implementations

CoCoA free computer algebra system for computing Gröbner bases.

GAP free computer algebra system that can perform Gröbner bases calculations.

FGb, Faugère's own implementation of his F4 algorithm, available as a Maple library.[5] To the
date, as of 2014, it is, with Magma, the fastest implementation for rational coefficients and
coefficients in a finite field of prime order

Macaulay 2 free software for doing polynomial computations, particularly Gröbner bases
calculations.

Magma has a very fast implementation of the Faugère's F4 algorithm.[6]

Maple has implementations of the Buchberger and Faugère F4 algorithms, as well as Gröbner
trace

Mathematica includes an implementation of the Buchberger algorithm, with performanceimproving techniques such as the Gröbner walk, Gröbner trace, and an improvement for toric bases

SINGULAR free software for computing Gröbner bases
AIRO INTERNATIONAL JOURNAL VOLUME 6 ISSN 23203714

Sage provides a unified interface to several computer algebra systems (including SINGULAR and
Macaulay), and includes a few Gröbner basis algorithms of its own.

SymPy Python computer algebra system uses Gröbner bases to solve polynomial systems
Coding theory plays an important role in efficient transmission of data over noisy communication
channels.

Based on experimentation and comparison of timings with other methods reported, we state a
tentative conclusion. The methods of this paper are viable and effective when the problem at
hand is unperturbed from an exactly solvable variant. They often give good results when the
problem is overdetermined, provided the noise is modest relative to an exactly solvable nearby
problem, and the scale of coefficients does not vary too much. In other situations it is not clear
whether our methods can be adapted so readily.

While most examples covered seem to work efficiently and give reasonable results, it remains
an open question as to how competitive these methods are in regard to speed and quality of
results, as compared to other approaches. An advantage to Gröbner bases is that polynomial
algebra is carried out in a sparse setting; many methods based on linear algebra require dense
matrix manipulation. The examples presented offer evidence that, when working with input of
modest degree, Gröbner bases methods are viable. That the coding is simple makes them all the
more attractive.
Hypothesis
The Gröbner bases method is a powerful tool in symbolic and algebraic computing, which is currently
not yet fully utilized in SymPy. Also implementation of Buchberger’s algorithm is quite limited at the
moment. However, as we showed in this chapter, SymPy can be used for solving practical problems in
symbolic mathematics, specifically problems which involve solving systems of polynomials. We hope
that, in foreseeable future, improved algorithms for computing Gröbner bases will be implemented, so
that SymPy will be able to tackle more complex problems.
Summary
AIRO INTERNATIONAL JOURNAL VOLUME 6 ISSN 23203714
In particular, the practical approach to find a minimal Gröbner basis is to calculate an
interreduced Gröbner basis. Interreduction means that all generators are in normal
form w.r.t. the rest of the generators. This is clearly not true here: x2y+x+1 can be
reduced by succesive reduction with x−y into y3+y+1, and xy2+y+1 into y3+y+1 as well.
The sets {x2y+x+1,xy2+y+1,x−y} and {y3+y+1,x−y} generate the same ideal (you can
write the elemtents of one in terms of the elements of the others), and it is
interreduced: y3+y+1 does not reduce x−y, and x−y does not reduce y3+y+1. Thus, this is
an interreduced Gröbner basis, and it is easy to see that it is also a minimal Gröbner
basisWe presented a new method for SVA properties checking by using Groebner bases based
symbolic algebraic approaches. To guarantee the feasibility we defined a constrained subset of
SVAs, which is powerful enough for practical purposes.
References
Adams, W. W. and Loustaunau, P. An Introduction to Gröbner Bases. Providence, RI: Amer.
Math. Soc., 1994.
Becker, T. and Weispfenning, V. Gröbner Bases: A Computational Approach to Commutative
Algebra. New York: Springer-Verlag, 1993.
Boege, W.; Gebauer, R.; and Kredel, H. "Some Examples for Solving Systems of Algebraic
Equations by Calculating Gröbner Bases." J. Symb. Comput. 1, 83-98, 1986.
Buchberger, B. "Gröbner Bases: An Algorithmic Method in Polynomial Ideal Theory." Ch. 6
in Multidimensional Systems Theory(Ed. N. K. Bose). New York: van Nostrand Reinhold, 1982.
Buchberger, B. "A Criterion for Detecting Unnecessary Reductions in the Construction of
Groebner Bases." Proceedings of the International Symposium on Symbolic and Algebraic
Computation. pp. 3-21, June 1979.
Buchberger,
B.
"Groebner
Bases:
A
Short
Introduction
for
Systems
Theorists." http://www.risc.uni-linz.ac.at/people/buchberg/papers/2001-02-19-A.pdf.
Buchberger, B. and Zapletal, A. "Gröbner Bases
Bibliography." http://www.ricam.oeaw.ac.at/Groebner-Bases-Bibliography/.
..
AIRO INTERNATIONAL JOURNAL VOLUME 6 ISSN 23203714
Cox, D.; Little, J.; and O'Shea, D. Ideals, Varieties, and Algorithms: An Introduction to
Algebraic Geometry and Commutative Algebra, 2nd ed. New York: Springer-Verlag, 1996.
Eisenbud, D. Commutative Algebra with a View toward Algebraic Geometry. New York:
Springer-Verlag, 1995.
Faugere, J. C.; Gianni, P.; Lazard, D.; and Mora, T. "Efficient Computation of Zero-Dimensional
Groebner Bases by Change of Ordering." J. Symb. Comput. 16, 329-344, 1993.
Giovini, A.; Mora, T.; Niesi, G.; Robbiano, L.; and Traverso, C. "One Sugar Cube, Please?, or
Selection Strategies in the Buchberger Algorithm." Proceedings of the International Symposium
on Symbolic and Algebraic Computation. pp. 49-54, June 1991.
Harris, J. "Rearranging Expressions by Patterns." Mathematica J.