The aim of this paper to draw attention to several aspects of the algebraic
dependence in algebras. The article starts with discussions of the algebraic
dependence problem in commutative algebras. Then the Burchnall-Chaundy
construction for proving algebraic dependence and obtaining the corresponding
algebraic curves for commuting differential operators in the Heisenberg algebra
is reviewed. Next some old and new results on algebraic dependence of commuting
q-difference operators and elements in q-deformed Heisenberg algebras are
reviewed. The main ideas and essence of two proofs of this are reviewed and
compared. One is the algorithmic dimension growth existence proof. The other is
the recent proof extending the Burchnall-Chaundy approach from differential
operators and the Heisenberg algebra to the q-deformed Heisenberg algebra,
showing that the Burchnall-Chaundy eliminant construction indeed provides
annihilating curves for commuting elements in the q-deformed Heisenberg
algebras for q not a root of unity.Comment: LaTeX, 14 pages, no figure
