I'm trying to write a program that solves sudoku's using a Groebner basis. I introduced 81 variables x₁ to x₈₁, this is a linearisation of the sudoku board.

The space of valid sudokus is defined by:

for i=1,…,81 : F_i=(x_i−1)(x_i−2)⋯(x_i−9) This represents the fact that all squares have integer values between 1 and 9.

for all x_i and x_j which...