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| GenOmega |
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This document briefly describes the GenOmega package written by Manuela Wiesinger.

The package mainly provides the functions Geno and GenoEq which apply the Omega>= and
Omega= operator, respectively, w.r.t. a certain variable to an expression.


The calling syntax is

                         Geno[A, B, U, t]   or   Geno[f, t]   or   Geno[Omega[f, t]]

and

                         GenoEq[A, B, U, t]    or    GenoEq[f, t]  or   GenoEq[OmegaEq[f, t]]


where t is a variable (i.e., a Mathematica symbol, an indexed symbol or a subscripted
symbol) and f is of the form

                               U(t)
                        -------------
                        A(t)   B(1/t)
with

U(t)     (=U)               a Laurent polynomial in t,
A(t), B(t)     (=A, B)      polynomials in t with A(1) != 0 and where for all roots xi of A and all roots yj of B
			    the constraint xi*yj != 1 holds.



Several variables can be eliminated  successively by calling (the input conditions have to hold in every elimination step!)

                         Geno[f,  {t1, ..., tr}]   or   Geno[Omega[f,   {t1, ..., tr}]]

and

                         GenoEq[f,  {t1, ..., tr}]   or   GenoEq[OmegaEq[f,  {t1, ..., tr}]]




Further information can be found at the GenOmega homepage

       -----------------------------------------------------------------------
      | http://www.risc.uni-linz.ac.at/research/combinat/risc/software/GenOmega/ |.
       -----------------------------------------------------------------------



Example:
-------

For applying GenOmega to (a special case of) Lecture Hall partitions
one proceeds as follows.


In[1]:= <<GenOmega.m          (* load the package *)

First we transform the inequalities into the corresponding Omega generating
function expression:


In[2]:= OSum[a1^n1 a2^n2 a3^n3 a4^n4, {3 n1 >= 4 n2, 2 n2 >= 3 n3, n3 >= 2 n4}, l]

        Assuming n1 >= 0
        Assuming n2 >= 0
        Assuming n3 >= 0
        Assuming n4 >= 0

                                    		          1
        Omega[-------------------------------------------------------------------------------------, {l_1 ,l_2 , l_3 }]
                   (1 - a1 (l_1)^3) (1 - a2 (l_2)^2/(l_1)^4) (1 - a4/(l_3)^2) (1 - a3 l_3/(l_2)^3)


Then we do the elimination:


In[3]:= Geno[%]

	

          1 + a1^2  a2 + a1^3  a2^2  + a1^3  a2^2  a3 + a1^4  a2^3  a3 + a1^6  a2^4  a3
Out[3]= ------------------------------------------------------------------------------------
          (-1 + a1) (-1 + a1^4  a2^3 ) (-1 + a1^4  a2^3  a3^2 ) (-1 + a1^4  a2^3  a3^2  a4)



=======================================================================
Manuela Wiesinger
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