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**Introduction to computational algebra**

**Valery Romanovski**

*Center for Applied Mathematics and Theoretical Physics,
*

University of Maribor, Maribor, Slovenia

Consider a system of polynomials

where are polynomials with coefficients from
some field (usually, is the field of real or complex numbers).
In algebraic geometry the solution space of system (1) is called
*the variety*. There are many numerical algorithms for solving
non-linear systems such as (1). These algorithms solve for one solution
at a time, and find an approximation to the solution.
They ignore the geometric properties of the solutions space (the
variety),
and do not take into consideration possible alternate descriptions of
the variety (using a different system of polynomials).
However recently efficient computational algorithms have been developed
which enable us to get algebraic and geometric information about
the *entire* solution space of system (1). They are based on the
Gröbner
bases theory worked out by B.Buchberger around the middle of 60th of
last century. The idea of the methods is to find the "best"
representation of the corresponding variety.
To illustrate this recall that the Gauss-Jordan elimination
method transforms a system of linear equations into the so-called
row echelon form. The system thus obtained has exactly the same
solutions
(the variety) as the original system, but it is trivial to solve.
The other example is the system (1) with being
polynomials in a single variable,
. In this case there is a polynomial
such that the system (1) is equivalent to the equation
. The polynomial is the greatest common divisor of
and it can be found using the Euclidean Algorithm.
Although the Gauss-Jordan elimination method cannot be directly
expanded to the case of non-linear polynomials and the Euclidean
Algorithm - to the case of
multivariable polynomials, it turns out that
an expansions of
these methods is possible, and, in a sense, the Gröbner bases theory
is a generalization of the Gauss-Jordan method and the Euclidean
Algorithm
to the case of non-linear multivariable polynomials.
In the lecture we give an introduction to the Gröbner bases theory,
discuss some algorithms implemented in computer algebra systems
(e.g. in Mathematica) and consider a few applications to the theory of
dynamics systems, in particular, to the investigation of the
time-reversible systems of ODE.

**References**

Adams W W and Loustaunau P 1994 *An Introduction to Gröbner
bases.* Graduate Studies in Mathematics.-V.3. RI:AMS.

Becker T and Weispfenning V 1993
*Gröbner Bases: A computational Approach to Commutative Algebra.*
Graduate Texts in Mathematics, V.141. Berlin and New York: Springer-Verlag.

Buchberger B 1985 in
*Multidimensional Systems Theory*, ed. N.K.Boese.
D.Reidel Pub. Co., 184-232.

Cox D, Little J, O'Shea D 1992 *Ideals, Varieties,
and Algorithms.* New York: Springer-Verlag.

Jarrah, A, Laubenbacher, R and Romanovski V 2001 The center variety of
polynomial differential systems, preprint,
`http://xxx.lanl.gov/abs/math.DS/0009061`

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