Computation in multivariate quaternionic polynomial ring (Q2871249)

Let \(\mathbb{H}\) denote the algebra of real quaternions. For a monomial well--ordering the concept of Gröbner bases for the (non--commutative) polynominal ring \(\mathbb {H}[x_1, \ldots, x_n]\) is explained. With \(a,b\in\mathbb{H}\) the multiplication is defined by \(ax_1^{\alpha_1}\cdot\ldots\cdot x_n^{\alpha_n}\ast bx_1^{\beta_1}\cdot\ldots\cdot x_n^{\beta_n}=abx_1^{\alpha_1+\beta_1}\cdot\ldots\cdot x_n^{\alpha_n+\beta_n}\).NEWLINENEWLINEIt is explained how to use the computer algebra system \texttt{Singular} to define \(\mathbb{H}[x_1, \ldots, x_n]\) and compute a left Gröbner basis for a given left ideal.