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Gessel walks are lattice walks in the quarter plane $\set N^2$ which start atthe origin $(0,0)\in\set N^2$ and consist only of steps chosen from the set$\{\leftarrow,\swarrow,\nearrow,\to\}$. We prove that if $g(n;i,j)$ denotes thenumber of Gessel walks of length $n$ which end at the point $(i,j)\in\set N^2$,then the trivariate generating series $G(t;x,y)=\sum_{n,i,j\geq 0} g(n;i,j)x^iy^j t^n$ is an algebraic function.

The complete Generating Function for Gessel Walks is Algebraic