---------------- | GenOmega | ---------------- 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]:= <= 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 =======================================================================