Wei Chen, Isomorphisms of Polynomial Objects and Algebraic Equations
Isomorphisms of Polynomial Objects and Algebraic Equations
Algebraic equations on complex numbers and functional equations on generating functions are often used to solve combinatorial problems. But the introduction of arithmetic operators such as subtraction and division always cause panic in the world of polynomial objects which are generated from constants by applying products and coproducts.
This research is to investigate connections between algebraic equations on complex numbers and isomorphisms of polynomial objects. We are attempting to work out conditions under which isomorphisms between polynomial objects can be decided by equalities between polynomials on multi-variables with integers as coefficients.
We start from Lawvere's Remark, also known as seven-trees-in-one, and illustrate it as a combinatorial one-person board game. An algorithm is given to solve these games. Base on this, following Fiore and Leinster's work, we extend their condition on isomorphisms of polynomial objects on a single variable to multi-variables. Some possible future work is discussed at the end.