调用 list() 时 Sage 中的 AttributeError

Question

I had some problem with error in Sage (I just cannot find a way to solve it)

Here is a code

nb = 8
varl = [ c + str( p ) for c in 'xyz' for p in range(nb)]
B = BooleanPolynomialRing ( names = varl )
B.inject_variables ()
P.<p> = PolynomialRing ( B )
Byte.<t> = P.quotient_ring ( p^8 + p^4 + p^3 + p + 1)
X = B.gens()[: nb ]
Y = B.gens()[ nb :2*nb ]
x = sum ([ X [ j ]*t^j for j in range(nb)])
y = sum ([ Y [ j ]*t^j for j in range(nb)])

E3 = x*y
ep3 = E3.list()
latex(ep3)

and output

Defining x0, x1, x2, x3, x4, x5, x6, x7, y0, y1, y2, y3, y4, y5, y6, y7, z0, z1, z2, z3, z4, z5, z6, z7
Error in lines 12-12
Traceback (most recent call last):
  File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 1188, in execute
    flags=compile_flags) in namespace, locals
  File "", line 1, in <module>
  File "sage/structure/element.pyx", line 489, in sage.structure.element.Element.__getattr__ (build/cythonized/sage/structure/element.c:4611)
    return self.getattr_from_category(name)
  File "sage/structure/element.pyx", line 502, in sage.structure.element.Element.getattr_from_category (build/cythonized/sage/structure/element.c:4720)
    return getattr_from_other_class(self, cls, name)
  File "sage/cpython/getattr.pyx", line 389, in sage.cpython.getattr.getattr_from_other_class (build/cythonized/sage/cpython/getattr.c:2540)
    raise AttributeError(dummy_error_message)
AttributeError: 'QuotientRing_generic_with_category.element_class' object has no attribute 'list'

I don't understand, why it not working. I tried to make it in cocalc.com and also in desktop version of Sage, but it didn't work.

Answer 1

Unfortunately, E3.list() doesn't work. Maybe it used to in Sage 6.7 when the cited paper came out, but it doesn't now. The following should work: keep everything you have up to and including E3 = x*y . At this point, E3 is

x7*y7*t^14 + (x6*y7 + x7*y6)*t^13 + (x5*y7 + x6*y6 + x7*y5)*t^12 + (x4*y7 + x5*y6 + x6*y5 + x7*y4)*t^11 + (x3*y7 + x4*y6 + x5*y5 + x6*y4 + x7*y3)*t^10 + (x2*y7 + x3*y6 + x4*y5 + x5*y4 + x6*y3 + x7*y2)*t^9 + (x1*y7 + x2*y6 + x3*y5 + x4*y4 + x5*y3 + x6*y2 + x7*y1)*t^8 + (x0*y7 + x1*y6 + x2*y5 + x3*y4 + x4*y3 + x5*y2 + x6*y1 + x7*y0)*t^7 + (x0*y6 + x1*y5 + x2*y4 + x3*y3 + x4*y2 + x5*y1 + x6*y0)*t^6 + (x0*y5 + x1*y4 + x2*y3 + x3*y2 + x4*y1 + x5*y0)*t^5 + (x0*y4 + x1*y3 + x2*y2 + x3*y1 + x4*y0)*t^4 + (x0*y3 + x1*y2 + x2*y1 + x3*y0)*t^3 + (x0*y2 + x1*y1 + x2*y0)*t^2 + (x0*y1 + x1*y0)*t + x0*y0

So then do:

E3L = E3.lift()
[E3L.monomial_coefficient(_) for _ in E3L.monomials()]

This returns

[x7*y7,
 x6*y7 + x7*y6,
 x5*y7 + x6*y6 + x7*y5,
 x4*y7 + x5*y6 + x6*y5 + x7*y4,
 x3*y7 + x4*y6 + x5*y5 + x6*y4 + x7*y3,
 x2*y7 + x3*y6 + x4*y5 + x5*y4 + x6*y3 + x7*y2,
 x1*y7 + x2*y6 + x3*y5 + x4*y4 + x5*y3 + x6*y2 + x7*y1,
 x0*y7 + x1*y6 + x2*y5 + x3*y4 + x4*y3 + x5*y2 + x6*y1 + x7*y0,
 x0*y6 + x1*y5 + x2*y4 + x3*y3 + x4*y2 + x5*y1 + x6*y0,
 x0*y5 + x1*y4 + x2*y3 + x3*y2 + x4*y1 + x5*y0,
 x0*y4 + x1*y3 + x2*y2 + x3*y1 + x4*y0,
 x0*y3 + x1*y2 + x2*y1 + x3*y0,
 x0*y2 + x1*y1 + x2*y0,
 x0*y1 + x1*y0,
 x0*y0]

You could also do latex(...) applied to the whole thing, or you could do

[latex(E3L.monomial_coefficient(_)) for _ in E3L.monomials()]

to get the list of LaTeX expressions for the coefficients.

Alternatively:

E3L = E3.lift()
E3L.list()

produces the list of coefficients but in increasing order of degree rather than decreasing: it returns

[x0*y0,
 x0*y1 + x1*y0,
 x0*y2 + x1*y1 + x2*y0,
 x0*y3 + x1*y2 + x2*y1 + x3*y0,
 x0*y4 + x1*y3 + x2*y2 + x3*y1 + x4*y0,
 x0*y5 + x1*y4 + x2*y3 + x3*y2 + x4*y1 + x5*y0,
 x0*y6 + x1*y5 + x2*y4 + x3*y3 + x4*y2 + x5*y1 + x6*y0,
 x0*y7 + x1*y6 + x2*y5 + x3*y4 + x4*y3 + x5*y2 + x6*y1 + x7*y0,
 x1*y7 + x2*y6 + x3*y5 + x4*y4 + x5*y3 + x6*y2 + x7*y1,
 x2*y7 + x3*y6 + x4*y5 + x5*y4 + x6*y3 + x7*y2,
 x3*y7 + x4*y6 + x5*y5 + x6*y4 + x7*y3,
 x4*y7 + x5*y6 + x6*y5 + x7*y4,
 x5*y7 + x6*y6 + x7*y5,
 x6*y7 + x7*y6,
 x7*y7]