Inverse for the shuffle for algebraic series

摘要

Let K be a commutative field of characteristic 0, and a an algebraic series of one variable with coefficients in K, which is invertible for the shuffle product, i.e. it exists an algebraic series b over K such that a̫b==1. Then a is a rational series. The same result is true when a and b are an algebraic series of many non-commutative variables. One application of this result is: If M is a linear system which is invertible, and with an output function ) and and N is a linear system with an output function ) suc such that N and the gen and the generating series H for M and G for N are both algebraic, then the systems M and N are bilinear.