Goss polynomials#
Goss polynomials for the Carlitz module
AUTHORS:
David Ayotte (2022): initial version
- drinfeld_modular_forms.goss_polynomials.bracket(n, polynomial_ring)#
Return the element \([n] = T^{q^n} - T\) where \(T\) is the generator of the polynomial ring.
EXAMPLES:
sage: from drinfeld_modular_forms import bracket sage: A.<T> = GF(3)['T'] sage: bracket(1, A) T^3 + 2*T sage: bracket(2, A) T^9 + 2*T sage: bracket(0, A) Traceback (most recent call last): ... ValueError: the integer n (=0) must be postive.
- drinfeld_modular_forms.goss_polynomials.goss_polynomial(n, polynomial_ring)#
Return the \(n\)-th Goss polynomial for the Carlitz module.
EXAMPLES:
sage: from drinfeld_modular_forms import goss_polynomial sage: A.<T> = GF(3)['T'] sage: goss_polynomial(1, A) X sage: goss_polynomial(2, A) X^2 sage: goss_polynomial(3, A) X^3 sage: goss_polynomial(4, A) X^4 + (1/(T^3 + 2*T))*X^2 sage: goss_polynomial(5, A) X^5 + (2/(T^3 + 2*T))*X^3 sage: goss_polynomial(6, A) X^6
- drinfeld_modular_forms.goss_polynomials.lcm_of_monic_polynomials(n, polynomial_ring)#
Return the least common multiple of all monic polynomials of degree \(n\).
EXAMPLES:
sage: from drinfeld_modular_forms import lcm_of_monic_polynomials sage: A.<T> = GF(3)['T'] sage: lcm_of_monic_polynomials(1, A) T^3 + 2*T sage: lcm_of_monic_polynomials(2, A) T^12 + 2*T^10 + 2*T^4 + T^2 sage: lcm_of_monic_polynomials(3, A) T^39 + 2*T^37 + 2*T^31 + T^29 + 2*T^13 + T^11 + T^5 + 2*T^3
- drinfeld_modular_forms.goss_polynomials.product_of_monic_polynomials(n, polynomial_ring)#
Return the product of all monic polynomials of degree \(n\).
EXAMPLES:
sage: from drinfeld_modular_forms import product_of_monic_polynomials sage: A.<T> = GF(3)['T'] sage: product_of_monic_polynomials(0, A) 1 sage: product_of_monic_polynomials(1, A) T^3 + 2*T sage: f = product_of_monic_polynomials(2, A); f T^18 + 2*T^12 + 2*T^10 + T^4 sage: f.factor() T^4 * (T + 1)^4 * (T + 2)^4 * (T^2 + 1) * (T^2 + T + 2) * (T^2 + 2*T + 2)