ore_algebra_t

The predefined Ore algebra structures, an enumeration with the values below. Except for the commutative and Frobenius algebras, they act on a distinguished generator $x$ of the base ring, which must currently be a univariate polynomial ring (created with gr_ctx_init_gr_poly); otherwise $\sigma$ and $\delta$ return GR_UNABLE.

• ORE_ALGEBRA_CUSTOM: $\sigma$ and $\delta$ given by a user function (gr_ore_poly_ctx_init_custom).
• ORE_ALGEBRA_COMMUTATIVE: $\sigma = \mathrm{id}$, $\delta = 0$; ordinary polynomials in $D$.
• ORE_ALGEBRA_DERIVATIVE: $\sigma = \mathrm{id}$, $\delta = d/dx$; linear differential operators.
• ORE_ALGEBRA_EULER_DERIVATIVE: $\sigma = \mathrm{id}$, $\delta = x \, d/dx$ (the Euler operator $\theta$).
• ORE_ALGEBRA_FORWARD_SHIFT: $\sigma(a)(x) = a(x + 1)$, $\delta = 0$; linear recurrence operators.
• ORE_ALGEBRA_FORWARD_DIFFERENCE: $\sigma(a)(x) = a(x + 1)$, $\delta(a) = a(x + 1) - a(x)$ (the operator $\Delta$).
• ORE_ALGEBRA_BACKWARD_SHIFT: $\sigma(a)(x) = a(x - 1)$, $\delta = 0$.
• ORE_ALGEBRA_BACKWARD_DIFFERENCE: $\sigma(a)(x) = a(x - 1)$, $\delta(a) = a(x) - a(x - 1)$ (the operator $\nabla$).
• ORE_ALGEBRA_Q_SHIFT: $\sigma(a)(x) = a(qx)$, $\delta = 0$; linear $q$-difference operators (gr_ore_poly_ctx_init_q_shift).
• ORE_ALGEBRA_MAHLER: $\sigma(a)(x) = a(x^b)$, $\delta = 0$; linear Mahler operators with base $b$ (gr_ore_poly_ctx_init_mahler).
• ORE_ALGEBRA_FROBENIUS: $\sigma$ the Frobenius automorphism $a \mapsto a^p$ of a finite field ($\delta = 0$), computed with gr_fq_frobenius.

The value ORE_POLY_NUM_ALGEBRAS is the number of algebra types.