InfinitePolynomialRing with coefficients in another InfinitePolynomialRing

[mantepse]

Problem Description

I frequently use polynomial rings over other polynomial rings for easy extraction of terms. For example:

sage: R.<a> = QQ[]
sage: S.<x, y, z> = R[]
sage: p = (x+y+z)^2*sum(a^k for k in range(4))
sage: p.monomial_coefficient(x*y)
2*a^3 + 2*a^2 + 2*a + 2

This does not work for infinite polynomial rings:

sage: R.<a> = InfinitePolynomialRing(QQ)
sage: S.<b> = InfinitePolynomialRing(R)
sage: p = (b[0] + b[1] + b[2])^2*sum(a[k] for k in range(4))
sage: p.monomial_coefficient(b[2]*b[1])
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
Cell In [132], line 1
----> 1 p.monomial_coefficient(b[Integer(2)]*b[Integer(1)])

TypeError: Argument 'mon' has incorrect type (expected sage.rings.polynomial.multi_polynomial_libsingular.MPolynomial_libsingular, got InfinitePolynomial_dense)

Proposed Solution

Get someone to fix the InfinitePolynomialRing.

Alternatives Considered

Do it myself, but upon consideration, this is not a viable solution.

Additional Information

No response

Is there an existing issue for this?

• I have searched the existing issues for a bug report that matches the one I want to file, without success.

[RuchitJagodara]

Get someone to fix the InfinitePolynomialRing.

Can you explain me what do you men by fixing the InfinitePolynomialRing ?

I have also looked into the working of this and I didn't find any problem with InfinitePolynomialRing, instead from the error message,

TypeError Traceback (most recent call last)
Cell In [132], line 1
----> 1 p.monomial_coefficient(b[Integer(2)]*b[Integer(1)])

TypeError: Argument 'mon' has incorrect type (expected >sage.rings.polynomial.multi_polynomial_libsingular.MPolynomial_libsingular, got InfinitePolynomial_dense)

It is clear that there is an issue with monomial_coefficient() function, when I checked what is the exact issue then I found that it is taking argument mon of a particular type only (i.e. MPolynomial_libsingular), and here when we are defining an InfinitePolynomialRing then we are passing different type of argument in the function so I think we need to make a new function/files specifically for InfinitePolynomialRings because multi_polynomial_libsingular.pyx does not seem to include functions related to infinite polynomial rings.

If I am wrong somewhere, @mantepse please tell me.

[RuchitJagodara]

@mantepse ?

[mantepse]

Get someone to fix the InfinitePolynomialRing.

Can you explain me what do you mean by fixing the InfinitePolynomialRing ?

There are several problems, for example #36576, also that InfinitePolynomialRing.gens is abused, etc.

I don't have any good answers, unfortunately.

[tdupu]

I found a related problem in the way that it PolynomialRing works over InfinitePolynomialRing handles adjoining ordinary variables:

A.<x,y> = InfinitePolynomialRing(QQ, order='deglex')
R.<t>=PolynomialRing(A,1)
I=R.ideal(t^2)
Rquo=R.quo(I)
#Rquo(x[1]) #this doesn't work.
z=R(x[1]) #this is fine
Rquo(z) #this freaks out.

Error in lines 6-6
Traceback (most recent call last):
  File "sage/misc/cachefunc.pyx", line 1962, in sage.misc.cachefunc.CachedMethodCaller.__call__
    return cache[k]
KeyError: (('', None, None, False), ())

During handling of the above exception, another exception occurred:

Traceback (most recent call last):
  File "/ext/sage/10.3/src/sage/rings/polynomial/multi_polynomial_ideal.py", line 4671, in groebner_basis
    gb = self._groebner_basis_libsingular("groebner", deg_bound=deg_bound, mult_bound=mult_bound, *args, **kwds)
         ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
  File "/ext/sage/10.3/src/sage/libs/singular/standard_options.py", line 143, in wrapper
    return func(*args, **kwds)
           ^^^^^^^^^^^^^^^^^^^
  File "/ext/sage/10.3/src/sage/rings/polynomial/multi_polynomial_ideal.py", line 554, in _groebner_basis_libsingular
    S = groebner(self)
        ^^^^^^^^^^^^^^
  File "sage/libs/singular/function.pyx", line 1297, in sage.libs.singular.function.SingularFunction.__call__
    raise TypeError("cannot call Singular function '%s' with ring parameter of type '%s'" % (self._name,type(ring)))
TypeError: cannot call Singular function 'groebner' with ring parameter of type '<class 'sage.rings.polynomial.multi_polynomial_ring.MPolynomialRing_polydict_with_category'>'

During handling of the above exception, another exception occurred:

Traceback (most recent call last):
  File "sage/misc/cachefunc.pyx", line 1962, in sage.misc.cachefunc.CachedMethodCaller.__call__
    return cache[k]
KeyError: (('groebner', Singular), (('deg_bound', None), ('mult_bound', None)))

During handling of the above exception, another exception occurred:

Traceback (most recent call last):
  File "sage/structure/category_object.pyx", line 855, in sage.structure.category_object.CategoryObject.getattr_from_category
    return self._cached_methods[name]
KeyError: '_PolynomialRing_singular_repr__singular'

During handling of the above exception, another exception occurred:

Traceback (most recent call last):
  File "/ext/sage/10.3/src/sage/rings/polynomial/polynomial_singular_interface.py", line 335, in _singular_
    R = self.__singular
        ^^^^^^^^^^^^^^^
  File "sage/structure/category_object.pyx", line 849, in sage.structure.category_object.CategoryObject.__getattr__
    return self.getattr_from_category(name)
  File "sage/structure/category_object.pyx", line 864, in sage.structure.category_object.CategoryObject.getattr_from_category
    attr = getattr_from_other_class(self, cls, name)
  File "sage/cpython/getattr.pyx", line 357, in sage.cpython.getattr.getattr_from_other_class
    raise AttributeError(dummy_error_message)
AttributeError: 'Singular' object has no attribute '_strip_prompt'

During handling of the above exception, another exception occurred:

Traceback (most recent call last):
  File "/ext/sage/10.3/src/sage/rings/polynomial/multi_polynomial_ideal.py", line 4674, in groebner_basis
    gb = self._groebner_basis_singular("groebner", deg_bound=deg_bound, mult_bound=mult_bound, *args, **kwds)
         ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
  File "/ext/sage/10.3/src/sage/interfaces/singular.py", line 2805, in wrapper
    return func(*args, **kwds)
           ^^^^^^^^^^^^^^^^^^^
  File "/ext/sage/10.3/src/sage/rings/polynomial/multi_polynomial_ideal.py", line 1480, in _groebner_basis_singular
    S = self._groebner_basis_singular_raw(algorithm=algorithm, *args, **kwds)
        ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
  File "sage/misc/cachefunc.pyx", line 1967, in sage.misc.cachefunc.CachedMethodCaller.__call__
    w = self._instance_call(*args, **kwds)
  File "sage/misc/cachefunc.pyx", line 1842, in sage.misc.cachefunc.CachedMethodCaller._instance_call
    return self.f(self._instance, *args, **kwds)
  File "/ext/sage/10.3/src/sage/rings/polynomial/multi_polynomial_ideal.py", line 1515, in _groebner_basis_singular_raw
    S = self._singular_()   # for degBound, we need to ensure
        ^^^^^^^^^^^^^^^^^
  File "/ext/sage/10.3/src/sage/rings/polynomial/multi_polynomial_ideal.py", line 633, in _singular_
    self.ring()._singular_(singular).set_ring()
    ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
  File "/ext/sage/10.3/src/sage/rings/polynomial/polynomial_singular_interface.py", line 349, in _singular_
    return self._singular_init_(singular)
           ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
  File "/ext/sage/10.3/src/sage/rings/polynomial/polynomial_singular_interface.py", line 366, in _singular_init_
    raise TypeError("no conversion of this ring to a Singular ring defined")
TypeError: no conversion of this ring to a Singular ring defined

During handling of the above exception, another exception occurred:

Traceback (most recent call last):
  File "/cocalc/lib/python3.11/site-packages/smc_sagews/sage_server.py", line 1244, in execute
    exec(
  File "", line 1, in <module>
  File "sage/structure/parent.pyx", line 901, in sage.structure.parent.Parent.__call__
    return mor._call_(x)
  File "sage/structure/coerce_maps.pyx", line 163, in sage.structure.coerce_maps.DefaultConvertMap_unique._call_
    raise
  File "sage/structure/coerce_maps.pyx", line 158, in sage.structure.coerce_maps.DefaultConvertMap_unique._call_
    return C._element_constructor(x)
  File "/ext/sage/10.3/src/sage/rings/quotient_ring.py", line 1084, in _element_constructor_
    return self.element_class(self, x)
           ^^^^^^^^^^^^^^^^^^^^^^^^^^^
  File "/ext/sage/10.3/src/sage/rings/quotient_ring_element.py", line 101, in __init__
    self._reduce_()
  File "/ext/sage/10.3/src/sage/rings/quotient_ring_element.py", line 120, in _reduce_
    self.__rep = I.reduce(self.__rep)
                 ^^^^^^^^^^^^^^^^^^^^
  File "/ext/sage/10.3/src/sage/rings/polynomial/multi_polynomial_ideal.py", line 4965, in reduce
    gb = self.groebner_basis()
         ^^^^^^^^^^^^^^^^^^^^^
  File "sage/misc/cachefunc.pyx", line 1967, in sage.misc.cachefunc.CachedMethodCaller.__call__
    w = self._instance_call(*args, **kwds)
  File "sage/misc/cachefunc.pyx", line 1842, in sage.misc.cachefunc.CachedMethodCaller._instance_call
    return self.f(self._instance, *args, **kwds)
  File "/ext/sage/10.3/src/sage/rings/qqbar_decorators.py", line 97, in wrapper
    return func(*args, **kwds)
           ^^^^^^^^^^^^^^^^^^^
  File "/ext/sage/10.3/src/sage/rings/polynomial/multi_polynomial_ideal.py", line 4706, in groebner_basis
    gb = toy_buchberger.buchberger_improved(self, *args, **kwds)
         ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
  File "/ext/sage/10.3/src/sage/rings/polynomial/toy_buchberger.py", line 260, in buchberger_improved
    F = inter_reduction(F.gens())
        ^^^^^^^^^^^^^^^^^^^^^^^^^
  File "/ext/sage/10.3/src/sage/rings/polynomial/toy_buchberger.py", line 442, in inter_reduction
    h = p.reduce(Q)
        ^^^^^^^^^^^
  File "/ext/sage/10.3/src/sage/rings/polynomial/multi_polynomial_element.py", line 2433, in reduce
    raise TypeError("Can only reduce polynomials over fields.")
TypeError: Can only reduce polynomials over fields.

In general I found it difficult to define ring homomorphisms and I couldn't quite figure out what the ring homomorphism issue traced back to. My goal was to define a derivation by using a ring of dual numbers A[t]/(t^2) and then specify the homomorphism by what happens on the generators. The way that this library has a special class for generators seems to have something to do with this.

This was run on cocalc on July 15 2024.
'SageMath version 10.3, Release Date: 2024-03-19'