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Authors: Harald Schilly, William A. Stein
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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]

Abelfunctions

by Chris Swierczewski

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()