**Franz Pauer
Institut fuer Mathematik, Universitaet Innsbruck
A-6020 Innsbruck, Technikerstr. 25/7, Austria
Innsbruck, A-6020
Austria
E-mail:Franz.Pauer@uibk.ac.at**

**Abstract**

Let *K* be a field and
the *K*-algebra of Laurent
polynomials over *K*. Let
be elements of *R* and
the set of their common zeros.

Clearly, there are elements
such
that
are polynomials in
and
.
Hence *Z* could be
computed via a Gröbner basis (with respect to a pure lexicographic
order) of the ideal generated by
in
.
But the degrees of these polynomials could be very
high, hence in this talk I shall propose another method to solve Laurent
polynomial equations. This method is based on results of
the paper PU (Pauer, F., Unterkircher, A.: Gröbner
Bases for Ideals in Laurent polynomial rings and their Application to
Systems of Difference Equations. Preprint 1997,
submitted to AAECC).

Let
,
let *T*_{0} be the submonoid of *T* generated by
and *T*_{j} the
submonoid generated by
.

Let
and let
<_{lex} be the lexicographic order on .
Then we define a
total order < on *T* as follows:

Then < is a

1 is the smallest element in *T* and

*r*<*s* implies *tr*<*ts*, for all

Moreover, all elements of *T*_{0} are smaller then any element
of
.

By *lt*(*f*) we denote the leading term (with respect to <)
of a non-zero Laurent polynomial .

Definition: Let *J* be an ideal in *R* and let *G*be a finite subset of
.
Then *G* is a *Gröbner basis* (with respect to <) of *J* iff

(Since < is not a term order, in general
!)

Proposition (see PU): For every ideal in *R* a
Gröbner basis can be computed in a finite number of steps.

For the sake of simlicity let us assume that *Z* is
finite. Let *J* be the ideal in *R* generated by the Laurent
polynomials
and let *x*_{n} be the smallest
element in
.
Then the ideal
in
is not zero. To get an
algorithm for the computation of *Z* it is sufficient to know how to
compute a finite system of generators of this ideal. Similar to the
well-known case of polynomial ideals we obtain the following

Proposition: Let *G* be a Gröbner basis of *J* with
respect to <. For
let *a*(*g*) be the
maximal nonnegative integer such that
*x*_{n}^{a(g)} divides *g* (in
). Then
generates the
ideal
.

IMACS ACA'98 Electronic Proceedings