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PIP packages for SageMath on SMC

Bases of Multivariate polynomials

by Viviane Pons

from multipolynomial_bases import *
A.<x> = MultivariatePolynomialAlgebra(QQ)
A
The Multivariate polynomial algebra on x over Rational Field 
2*x + 3*x +1
Ideal (2*x[0], 3*x[0], x[0]) of The Multivariate polynomial algebra on x over Rational Field on the monomial basis 
from multipolynomial_bases import DemazureHatPolynomials
HatDem = DemazureHatPolynomials(QQ)
HatDem
The Multivariate polynomial algebra on x over Rational Field on the Demazure hat basis of type A 
HatDem.an_element()
2*^K[1, 0, 0] + ^K[2, 2, 3] + ^K[0, 0, 0] + 3*^K[0, 1, 0] 
HatDem[1,2,2] + HatDem[2,3,1]
^K[1, 2, 2] + ^K[2, 3, 1] 
pol = HatDem[3,2,3] + HatDem[3,1,1]
pol.divided_difference(1)
^K[1, 2, 1] + ^K[2, 2, 3] + ^K[2, 1, 1] 
pol.isobaric_divided_difference(1)
^K[3, 2, 3] + ^K[1, 3, 1] + ^K[2, 3, 3] + ^K[3, 1, 1] 
from multipolynomial_bases import SchubertPolynomials
Schub = SchubertPolynomials(QQ)
HatDem(Schub([1,0,2]))
^K[3, 0, 0] + ^K[1, 0, 2] + ^K[2, 1, 0] + ^K[1, 2, 0] + ^K[2, 0, 1] 
from multipolynomial_bases import DemazurePolynomials
Dem = DemazurePolynomials(QQ)
Dem
The Multivariate polynomial algebra on x over Rational Field on the Demazure basis of type A 
Dem.an_element()
2*K[1, 0, 0] + K[2, 2, 3] + K[0, 0, 0] + 3*K[0, 1, 0] 
Dem[1,2,2] + Dem[2,3,1]
K[1, 2, 2] + K[2, 3, 1] 
from abelfunctions import *
R.<x,y> = QQ[]
f = y**3 + 2*x**3*y - x**7
X = RiemannSurface(f)
X
Riemann surface defined by f = -x^7 + 2*x^3*y + y^3 
X.genus()
2 
differentials = X.holomorphic_differentials()
for omega in differentials:
    print omega
x*y/(2*x^3 + 3*y^2) x^3/(2*x^3 + 3*y^2) 
a_cycles = X.a_cycles()
b_cycles = X.b_cycles()
xfig = a_cycles[0].plot_x(512)
yfig = a_cycles[0].plot_y(512, color='green')
xfig.show(); yfig.show()