Row-reduction and invariants of diophantine equations (Q1338153)

The author is interested in rational solutions with \(X_ 1 \dots X_ n \neq 0\) of the polynomial equation \(f(X_ 1, \dots, X_ n) = 0\) where the coefficients of \(f\) are integers. An integral matrix of exponents is associated to \(f\). By a multiplicative change of variables the matrix is brought to row-echelon form, which corresponds to an equation that is equivalent to the original but which sometimes can be more easily solved (e.g., \(f\) has degree 1 in a new variable). This method is also used for equations over finite fields.