## The complete generating function for Gessel walks is algebraic

### Supplementary Material

This is a collection of files accompanying an article of Alin Bostan and Manuel Kauers containing a proof that the complete generating function for Gessel walks is algebraic. The article itself can be downloaded here: [ps.gz] [pdf]

#### Section 1

• Maple Session [mpl]

#### Section 2

• Maple Session [mpl]
• Coefficients of F(t,x) mod t101 [mpl] [mma]
• The minimal polynomial of F(t,x) [mpl] [mma]

#### Section 3.1

• Coefficients of G(t,x,0) mod t1001 [mpl] [mma] (75Mb)
• Guessed differential operator for G(t,x,0) [mpl] [mma]
• The minimal polynomial of G(t,x,0) [mpl] [mma]
• Coefficients of G(t,0,y) mod t1001 [mpl] [mma] (45Mb)
• Guessed differential operator for G(t,0,y) [mpl] [mma]
• The minimal polynomial of G(t,0,y) [mpl] [mma]
• Coefficients of U(t,x) mod t1001 [mpl] [mma] (75Mb)
• The minimal polynomial of U(t,x) [mpl] [mma]
• Coefficients of V(t,x) mod t1001 [mpl] [mma] (45Mb)
• The minimal polynomial of V(t,x) [mpl] [mma]

#### Section 3.2

• Details on existence of Ucand(t,x) and Vcand(t,x) [ps] [pdf]
• Differential operators annihilating Ucand(t,x) [mma]
• Recurrence operators annihilating the coefficients of Ucand(t,x) [mma]
• Transformation matrix for Ucand(t,x) [mma]
• Witness operators for Ucand(t,x) [mma]
• The polynomial Q(T,t,y) whose solution is the auxiliary series f(t,y) [mma]
• Differential operators annihilating f(t,x) [mma]
• Recurrence operators annihilating the coefficients of f(t,x) [mma]
• First transformation matrix for f(t,x) [mma]
• First set of witness operators for f(t,x) [mma] (10Mb)
• Second transformation matrix for f(t,x) [mma]
• Second set of witness operators for f(t,x) [mma] (18Mb)

#### Section 3.3

• The minimal polynomial of (1+x)G2(t,x) - G(t;0,0) and x X(t,x)/t-G1(t,X(t,x)) [mpl] [mma]

#### Section 3.4

• Radical expression for G(t,1,1) [mpl] [mma]
• Minimal polynomial of G(t,1,1) [mpl] [mma]
• Radical expression for G(t,1,0) [mpl] [mma]
• Minimal polynomial of G(t,1,0) [mpl] [mma]
• Radical expression for G(t,0,1) [mpl] [mma]
• Minimal polynomial of G(t,0,1) [mpl] [mma]