.. _polys-domainsintro:

==========================================
Introducing the Domains of the poly module
==========================================

This page introduces the idea of the "domains" that are used in SymPy's
:mod:`sympy.polys` module. The emphasis is on introducing how to use the domains
directly and on understanding how they are used internally as part of the
:py:class:`~.Poly` class. This is a relatively advanced topic so for a more
introductory understanding of the :py:class:`~.Poly` class and the :mod:`sympy.polys`
module it is recommended to read :ref:`polys-basics` instead. The reference
documentation for the domain classes is in :ref:`polys-domainsref`. Internal
functions that make use of the domains are documented in
:ref:`polys-internals`.

What are the domains?
=====================

For most users the domains are only really noticeable in the printed output of
a :py:class:`~.Poly`::

  >>> from sympy import Symbol, Poly
  >>> x = Symbol('x')
  >>> Poly(x**2 + x)
  Poly(x**2 + x, x, domain='ZZ')
  >>> Poly(x**2 + x/2)
  Poly(x**2 + 1/2*x, x, domain='QQ')

We see here that one :py:class:`~.Poly` has domain :ref:`ZZ` representing the
integers and the other has domain :ref:`QQ` representing the rationals. These
indicate the "domain" from which the coefficients of the polynomial are drawn.

From a high-level the domains represent formal concepts such as the set of
integers `\mathbb{Z}` or rationals `\mathbb{Q}`. The word "domain" here is a
reference to the mathematical concept of an `integral domain`_.

.. _integral domain: https://en.wikipedia.org/wiki/Integral_domain

Internally the domains correspond to different computational implementations
and representations of the expressions that the polynomials correspond to.
The :py:class:`~.Poly` object itself has an internal representation as a
``list`` of coefficients and a :py:attr:`~.Poly.domain` attribute
representing the implementation of those coefficients::

  >>> p = Poly(x**2 + x/2)
  >>> p
  Poly(x**2 + 1/2*x, x, domain='QQ')
  >>> p.domain
  QQ
  >>> p.rep  # doctest: +SKIP
  DMP_Python([1, 1/2, 0], QQ)
  >>> p.rep.rep  # doctest: +SKIP
  [1, 1/2, 0]
  >>> type(p.rep.rep[0])  # doctest: +SKIP
  <class 'sympy.external.pythonmpq.PythonMPQ'>

Here the domain is :ref:`QQ` which represents the implementation of the
rational numbers in the domain system. The :py:class:`~.Poly` instance itself
has a :py:attr:`.Poly.domain` attribute :ref:`QQ` and then a list of
:py:class:`~.PythonMPQ` coefficients where :py:class:`~.PythonMPQ`
is the class that implements the elements of the :ref:`QQ` domain. The list of
coefficients ``[1, 1/2, 0]`` gives a standardised low-level representation of
the polynomial expression ``(1)*x**2 + (1/2)*x + (0)``.

This page looks at the different domains that are defined in SymPy, how they
are implemented and how they can be used. It introduces how to use the domains
and domain elements directly and explains how they are used internally as part
of :py:class:`~.Poly` objects. This information is more relevant for
development in SymPy than it is for users of the :mod:`sympy.polys` module.

Representing expressions symbolically
=====================================

There are many different ways that a mathematical expression can be
represented symbolically. The purpose of the polynomial domains is to provide
suitable implementations for different classes of expressions. This section
considers the basic approaches to the symbolic representation of mathematical
expressions: "tree", "dense polynomial"  and "sparse polynomial".

Tree representation
*******************

The most general representation of symbolic expressions is as a `tree`_ and
this is the representation used for most ordinary SymPy expressions which are
instances of :py:class:`~.Expr` (a subclass of :py:class:`~.Basic`). We can see
this representation using the :py:func:`~.srepr` function::

  >>> from sympy import Symbol, srepr
  >>> x = Symbol('x')
  >>> e = 1 + 1/(2 + x**2)
  >>> e
  1 + 1/(x**2 + 2)
  >>> print(srepr(e))
  Add(Integer(1), Pow(Add(Pow(Symbol('x'), Integer(2)), Integer(2)), Integer(-1)))

Here the expression ``e`` is represented as an :py:class:`~.Add` node which
has two children ``1`` and ``1/(x**2 + 2)``. The child ``1`` is represented as
an :py:class:`~.Integer` and the other child is represented as a :py:class:`~.Pow` with
base ``x**2 + 2`` and exponent ``1``. Then ``x**2 + 2`` is represented as an
:py:class:`~.Add` with children ``x**2`` and ``2`` and so on. In this way the
expression is represented as a tree where the internal nodes are operations
like :py:class:`~.Add`, :py:class:`~.Mul`, :py:class:`~.Pow` and so on and the
leaf nodes are atomic expression types like :py:class:`~.Integer` and
:py:class:`~.Symbol`. See :ref:`tutorial-manipulation` for more about this
representation.

The tree representation is core to the architecture of :py:class:`~.Expr` in
SymPy. It is a highly flexible representation that can represent a very wide
range of possible expressions. It can also represent equivalent expressions in
different ways e.g.::

  >>> e = x*(x + 1)
  >>> e
  x*(x + 1)
  >>> e.expand()
  x**2 + x

These two expression although equivalent have different tree representations::

  >>> print(srepr(e))
  Mul(Symbol('x'), Add(Symbol('x'), Integer(1)))
  >>> print(srepr(e.expand()))
  Add(Pow(Symbol('x'), Integer(2)), Symbol('x'))

Being able to represent the same expression in different ways is both a
strength and a weakness. It is useful to be able to convert an
expression in to different forms for different tasks but having non-unique
representations makes it hard to tell when two expressions are equivalent
which is in fact very important for many computational algorithms. The most
important task is being able to tell when an expression is equal to zero which
is undecidable in general (`Richardon's theorem`_) but is decidable in many
important special cases.

.. _tree: https://en.wikipedia.org/wiki/Tree_(data_structure)
.. _Richardon's theorem: https://en.wikipedia.org/wiki/Richardson%27s_theorem

.. _dup-representation:

DUP representation
******************

Restricting the set of allowed expressions to special cases allows for much
more efficient symbolic representations. As we already saw :py:class:`~.Poly`
can represent a polynomial as a list of coefficients. This means that an
expression like ``x**4 + x + 1`` could be represented simply as ``[1, 0, 0, 1,
1]``. This list of coefficients representation of a polynomial expression is
known as the "dense univariate polynomial" (DUP) representation. Working
within that representation algorithms for multiplication, addition and
crucially zero-testing can be much more efficient than with the corresponding
tree representations. We can see this representation from a :py:class:`~.Poly`
instance by looking it its ``rep.rep`` attribute::

  >>> p = Poly(x**4 + x + 1)
  >>> p.rep.rep  # doctest: +SKIP
  [1, 0, 0, 1, 1]

In the DUP representation it is not possible to represent the same expression
in different ways. There is no distinction between ``x*(x + 1)`` and ``x**2 +
x`` because both are just ``[1, 1, 0]``. This means that comparing two
expressions is easy: they are equal if and only if all of their coefficients
are equal. Zero-testing is particularly easy: the polynomial is zero if and
only if all coefficients are zero (of course we need to have easy zero-testing
for the coefficients themselves).

We can make functions that operate on the DUP representation much more
efficiently than functions that operate on the tree representation. Many
operations with standard sympy expressions are in fact computed by converting
to a polynomial representation and then performing the calculation. An example
is the :py:func:`~.factor` function::

  >>> from sympy import factor
  >>> e = 2*x**3 + 10*x**2 + 16*x + 8
  >>> e
  2*x**3 + 10*x**2 + 16*x + 8
  >>> factor(e)
  2*(x + 1)*(x + 2)**2

Internally :py:func:`~.factor` will convert the expression from the tree
representation into the DUP representation and then use the function
``dup_factor_list``::

  >>> from sympy import ZZ
  >>> from sympy.polys.factortools import dup_factor_list
  >>> p = [ZZ(2), ZZ(10), ZZ(16), ZZ(8)]
  >>> p
  [2, 10, 16, 8]
  >>> dup_factor_list(p, ZZ)
  (2, [([1, 1], 1), ([1, 2], 2)])

There are many more examples of functions with ``dup_*`` names for operating
on the DUP representation that are documented in :ref:`polys-internals`. There
are also functions with the ``dmp_*`` prefix for operating on multivariate
polynomials.

.. _dmp-representation:

DMP representation
******************

A multivariate polynomial (a polynomial in multiple variables) can be
represented as a polynomial with coefficients that are themselves polynomials.
For example ``x**2*y + x**2 + x*y + y + 1`` can be represented as polynomial
in ``x`` where the coefficients are themselves polynomials in ``y`` i.e.: ``(y
+ 1)*x**2 + (y)*x + (y+1)``. Since we can represent a polynomial with a list
of coefficients a multivariate polynomial can be represented with a list of
lists of coefficients::

  >>> from sympy import symbols
  >>> x, y = symbols('x, y')
  >>> p = Poly(x**2*y + x**2 + x*y + y + 1)
  >>> p
  Poly(x**2*y + x**2 + x*y + y + 1, x, y, domain='ZZ')
  >>> p.rep.rep  # doctest: +SKIP
  [[1, 1], [1, 0], [1, 1]]

This list of lists of (lists of...) coefficients representation is known as
the "dense multivariate polynomial" (DMP) representation.

.. _sparse-poly-representation:

Sparse polynomial representation
********************************

Instead of lists we can use a dict mapping nonzero monomial terms to their
coefficients. This is known as the "sparse polynomial" representation. We can
see what this would look like using the :py:meth:`~.Poly.as_dict` method::

  >>> Poly(7*x**20 + 8*x + 9).rep.rep  # doctest: +SKIP
  [7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 9]
  >>> Poly(7*x**20 + 8*x + 9).as_dict()
  {(0,): 9, (1,): 8, (20,): 7}

The keys of this dict are the exponents of the powers of ``x`` and the values
are the coefficients so e.g. ``7*x**20`` becomes ``(20,): 7`` in the dict. The
key is a tuple so that in the multivariate case something like ``4*x**2*y**3``
can be represented as ``(2, 3): 4``. The sparse representation can be more
efficient as it avoids the need to store and manipulate the zero coefficients.
With a large number of generators (variables) the dense representation becomes
particularly inefficient and it is better to use the sparse representation::

  >>> from sympy import prod
  >>> gens = symbols('x:10')
  >>> gens
  (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9)
  >>> p = Poly(prod(gens))
  >>> p
  Poly(x0*x1*x2*x3*x4*x5*x6*x7*x8*x9, x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, domain='ZZ')
  >>> p.rep.rep  # doctest: +SKIP
  [[[[[[[[[[1, 0], []], [[]]], [[[]]]], [[[[]]]]], [[[[[]]]]]], [[[[[[]]]]]]], [[[[[[[]]]]]]]], [[[[[[[[]]]]]]]]], [[[[[[[[[]]]]]]]]]]
  >>> p.as_dict()
  {(1, 1, 1, 1, 1, 1, 1, 1, 1, 1): 1}

The dict representation shown in the last output maps from the monomial which
is represented as a tuple of powers (``(1, 1, 1, ...)`` i.e. ``x0**1 * x1**1,
...``) to the coefficient ``1``. Compared to the :ref:`dmp-representation` we
have a much more flattened data structure: it is a ``dict`` with only one key
and value. Algorithms for working with sparse representations would likely be
much more efficient than dense algorithms for this particular example
polynomial.

SymPy's polynomial module has implementations of polynomial expressions based
on both the dense and sparse representations. There are also other
implementations of different special classes of expressions that can be used
as the coefficients of those polynomials. The rest of this page discusses what
those representations are and how to use them.

Basic usage of domains
======================

Several domains are predefined and ready to be used such as :ref:`ZZ` and :ref:`QQ`
which represent the ring of integers `\mathbb{Z}` and the field of rationals
`\mathbb{Q}`. The :py:class:`~.Domain` object is used to construct elements
which can then be used in ordinary arithmetic operations.::

  >>> from sympy import ZZ
  >>> z1 = ZZ(2)
  >>> z1
  2
  >>> z1 + z1
  4
  >>> type(z1)  # doctest: +SKIP
  <class 'int'>
  >>> z1 in ZZ
  True

The basic operations ``+``, ``-``, and ``*`` for addition, subtraction and
multiplication will work for the elements of any domain and will produce new
domain elements. Division with ``/`` (Python's "true division" operator) is not
possible for all domains and should not be used with domain elements unless
the domain is known to be a field. For example dividing two elements of :ref:`ZZ`
might give a ``float`` which is not an element of :ref:`ZZ`::

  >>> z1 / z1  # doctest: +SKIP
  1.0
  >>> type(z1 / z1)  # doctest: +SKIP
  <class 'float'>
  >>> ZZ.is_Field
  False

The behaviour of ``/`` for non-fields can also differ for different
implementations of the ground types of the domain. For example with
``SYMPY_GROUND_TYPES=flint`` dividing two elements of :ref:`ZZ` will raise an
error rather than return a float::

   >>> z1 / z1  # doctest: +SKIP
   Traceback (most recent call last):
   ...
   TypeError: unsupported operand type(s) for /: 'flint._flint.fmpz' and 'flint._flint.fmpz'

Most domains representing non-field rings allow floor and modulo division
(remainder) with Python's floor division ``//`` and modulo division ``%``
operators. For example with :ref:`ZZ`::

  >>> z1 // z1
  1
  >>> z1 % z1
  0

The :ref:`QQ` domain represents the field of rational numbers and does allow
division::

  >>> from sympy import QQ
  >>> q1 = QQ(1, 2)
  >>> q1
  1/2
  >>> q2 = QQ(2, 3)
  >>> q2
  2/3
  >>> q1 / q2
  3/4
  >>> type(q1)  # doctest: +SKIP
  <class 'sympy.external.pythonmpq.PythonMPQ'>

In general code that is expected to work with elements of an arbitrary domain
should not use the division operators ``/``, ``//`` and ``%``. Only the operators
``+``, ``-``, ``*`` and ``**`` (with nonnegative integer exponent) should be
assumed to work with arbitrary domain elements. All other operations should be
accessed as functions from the :py:class:`~.Domain` object::

  >>> ZZ.quo(ZZ(5), ZZ(3))  # 5 // 3
  1
  >>> ZZ.rem(ZZ(5), ZZ(3))  # 5 % 3
  2
  >>> ZZ.div(ZZ(5), ZZ(3))  # divmod(5, 3)
  (1, 2)
  >>> QQ.div(QQ(5), QQ(3))
  (5/3, 0)

The :py:meth:`~.Domain.exquo` function is used to compute an exact quotient.
This is the analogue of ``a / b`` but where the division is expected to be exact
(with no remainder) or an error will be raised::

  >>> QQ.exquo(QQ(5), QQ(3))
  5/3
  >>> ZZ.exquo(ZZ(4), ZZ(2))
  2
  >>> ZZ.exquo(ZZ(5), ZZ(3))
  Traceback (most recent call last):
  ...
  ExactQuotientFailed: 3 does not divide 5 in ZZ

The exact methods and attributes of the domain elements are not guaranteed in
general beyond the basic arithmetic operations. It should not be presumed that
e.g. :ref:`ZZ` will always be of type ``int``. If ``gmpy`` or ``gmpy2`` is
installed then the ``mpz`` or ``mpq`` types are used instead for :ref:`ZZ` and
:ref:`QQ`::

  >>> from sympy import ZZ, QQ
  >>> ZZ(2)  # doctest: +SKIP
  mpz(2)
  >>> QQ(2, 3)  # doctest: +SKIP
  mpq(2, 3)

The ``mpz`` type is faster than Python's standard ``int`` type for operations
with large integers although for smaller integers the difference is not so
significant. The ``mpq`` type representing rational numbers is implemented in
C rather than Python and is many times faster than the pure Python
implementation of :ref:`QQ` that is used when gmpy is not installed.

In general the Python type used for the elements of a domain can be checked
from the :py:attr:`~.Domain.dtype` attribute of the domain. When gmpy is
installed the dtype for :ref:`ZZ` is `mpz` which is not an actual type and can
not be used with `isinstance`. For this reason the :py:meth:`~.Domain.of_type`
method can be used to check if an object is an element of
:py:attr:`~.Domain.dtype`.::

  >>> z = ZZ(2)
  >>> type(z)  # doctest: +SKIP
  <class 'int'>
  >>> ZZ.dtype  # doctest: +SKIP
  <class 'int'>
  >>> ZZ.of_type(z)
  True

Domain elements vs sympy expressions
====================================

Note that domain elements are not of the same type as ordinary sympy
expressions which are subclasses of :py:class:`~.Expr` such as
:py:class:`~sympy.core.numbers.Integer`. Ordinary sympy expressions are
created with the :py:func:`~sympy.core.sympify.sympify` function.::

  >>> from sympy import sympify
  >>> z1_sympy = sympify(2)  # Normal sympy object
  >>> z1_sympy
  2
  >>> type(z1_sympy)
  <class 'sympy.core.numbers.Integer'>
  >>> from sympy import Expr
  >>> isinstance(z1_sympy, Expr)
  True

It is important when working with the domains not to mix sympy expressions
with domain elements even though it will sometimes work in simple cases. Each
domain object has the methods :py:meth:`~.Domain.to_sympy` and
:py:meth:`~.Domain.from_sympy` for converting back and forth between sympy
expressions and domain elements::

  >>> z_sympy = sympify(2)
  >>> z_zz = ZZ.from_sympy(z_sympy)
  >>> z_zz
  2
  >>> type(z_sympy)
  <class 'sympy.core.numbers.Integer'>
  >>> type(z_zz)  # doctest: +SKIP
  <class 'int'>
  >>> ZZ.to_sympy(z_zz)
  2
  >>> type(ZZ.to_sympy(z_zz))
  <class 'sympy.core.numbers.Integer'>

Any particular domain will only be able to represent some sympy expressions so
conversion will fail if the expression can not be represented in the domain::

  >>> from sympy import sqrt
  >>> e = sqrt(2)
  >>> e
  sqrt(2)
  >>> ZZ.from_sympy(e)
  Traceback (most recent call last):
  ...
  CoercionFailed: expected an integer, got sqrt(2)

We have already seen that in some cases we can use the domain object itself as
a constructor e.g. ``QQ(2)``. This will generally work provided the arguments
given are valid for the :py:attr:`~.Domain.dtype` of the domain. Although it is
convenient to use this in interactive sessions and in demonstrations it is
generally better to use the :py:meth:`~.Domain.from_sympy` method for
constructing domain elements from sympy expressions (or from objects that can
be sympified to sympy expressions).

It is important not to mix domain elements with other Python types such as
``int``, ``float``, as well as standard sympy :py:class:`~.Expr` expressions.
When working in a domain, care should be taken as some Python operations will
do this implicitly. for example the ``sum`` function will use the regular
``int`` value of zero so that ``sum([a, b])`` is effectively evaluated as ``(0
+ a) + b`` where ``0`` is of type ``int``.

Every domain is at least a ring if not a field and as such is guaranteed to
have two elements in particular corresponding to `1` and `0`.  The domain
object provides domain elements for these as the attributes
:py:attr:`~.Domain.one` and :py:attr:`~.Domain.zero`. These are useful for
something like Python's ``sum`` function which allows to provide an
alternative object as the "zero"::

  >>> ZZ.one
  1
  >>> ZZ.zero
  0
  >>> sum([ZZ(1), ZZ(2)])  # don't do this (even it sometimes works)
  3
  >>> sum([ZZ(1), ZZ(2)], ZZ.zero) # provide the zero from the domain
  3

A standard pattern then for performing calculations in a domain is:

#. Start with sympy :py:class:`~.Expr` instances representing expressions.
#. Choose an appropriate domain that can represent the expressions.
#. Convert all expressions to domain elements using
   :py:meth:`~.Domain.from_sympy`.
#. Perform the calculation with the domain elements.
#. Convert back to :py:class:`~.Expr` with :py:meth:`~.Domain.to_sympy`.

Here is an implementation of the ``sum`` function that illustrates these
steps and sums some integers but performs the calculation using the domain
elements rather than standard sympy expressions::

  def sum_domain(expressions_sympy):
      """Sum sympy expressions but performing calculations in domain ZZ"""

      # Convert to domain
      expressions_dom = [ZZ.from_sympy(e) for e in expressions_sympy]

      # Perform calculations in the domain
      result_dom = ZZ.zero
      for e_dom in expressions_dom:
          result_dom += e_dom

      # Convert the result back to Expr
      result_sympy = ZZ.to_sympy(result_dom)
      return result_sympy

Gaussian integers and Gaussian rationals
========================================

The two example domains that we have seen so far are :ref:`ZZ` and :ref:`QQ`
representing the integers and the rationals respectively. There are other
simple domains such as :ref:`ZZ_I` and :ref:`QQ_I` representing the
`Gaussian integers`_ and `Gaussian rationals`_. The Gaussian integers are
numbers of the form `a\sqrt{-1} + b` where `a` and `b` are integers. The
Gaussian rationals are defined similarly except that `a` and `b` can be
rationals. We can use the Gaussian domains like::

  >>> from sympy import ZZ_I, QQ_I, I
  >>> z = ZZ_I.from_sympy(1 + 2*I)
  >>> z
  (1 + 2*I)
  >>> z**2
  (-3 + 4*I)

Note the contrast with the way this calculation works in the tree
representation where :py:func:`~.expand` is needed to get the reduced form::

  >>> from sympy import expand, I
  >>> z = 1 + 2*I
  >>> z**2
  (1 + 2*I)**2
  >>> expand(z**2)
  -3 + 4*I

The :ref:`ZZ_I` and :ref:`QQ_I` domains are implemented by the classes
:py:class:`~.GaussianIntegerRing` and :py:class:`~.GaussianRationalField` and
their elements by :py:class:`~.GaussianInteger` and
:py:class:`~.GaussianRational` respectively. The internal representation for
an element of :ref:`ZZ_I` or :ref:`QQ_I` is simply as a pair ``(a, b)`` of
elements of :ref:`ZZ` or :ref:`QQ` respectively. The domain :ref:`ZZ_I` is a
ring with similar properties to :ref:`ZZ` whereas :ref:`QQ_I` is a field much
like :ref:`QQ`::

  >>> ZZ.is_Field
  False
  >>> QQ.is_Field
  True
  >>> ZZ_I.is_Field
  False
  >>> QQ_I.is_Field
  True

Since :ref:`QQ_I` is a field division by nonzero elements is always possible
whereas in :ref:`ZZ_I` we have the important concept of the greatest common
divisor (GCD)::

  >>> e1 = QQ_I.from_sympy(1+I)
  >>> e2 = QQ_I.from_sympy(2-I/2)
  >>> e1/e2
  (6/17 + 10/17*I)
  >>> ZZ_I.gcd(ZZ_I(5), ZZ_I.from_sympy(1+2*I))
  (1 + 2*I)

.. _Gaussian integers: https://en.wikipedia.org/wiki/Gaussian_integer
.. _Gaussian rationals: https://en.wikipedia.org/wiki/Gaussian_rational

Finite fields
=============

So far we have seen the domains :ref:`ZZ`, :ref:`QQ`, :ref:`ZZ_I`, and
:ref:`QQ_I`. There are also domains representing the `Finite fields`_ although
the implementation of these is incomplete. A finite field :ref:`GF(p)` of
*prime* order can be constructed with ``FF`` or ``GF``. A domain for the
finite field of prime order `p` can be constructed with :ref:`GF(p)`::

  >>> from sympy import GF
  >>> K = GF(5)
  >>> two = K(2)
  >>> two #doctest: +SKIP
  2 mod 5
  >>> two ** 2A #doctest: +SKIP
  4 mod 5
  >>> two ** 3 #doctest: +SKIP
  3 mod 5

There is also ``FF`` as an alias for ``GF`` (standing for "finite field" and
"Galois field" respectively). These are equivalent and both ``FF(n)`` and
``GF(n)`` will create a domain which is an instance of
:py:class:`~.FiniteField`. The associated domain elements will be instances of
:py:class:`~.PythonFiniteField` or :py:class:`~.GMPYFiniteField` depending on
whether or not ``gmpy`` is installed.

Finite fields of order `p^n` where `n \ne 1` are not implemented. It is
possible to use e.g. ``GF(6)`` or ``GF(9)`` but the resulting domain is *not*
a field. It is just the integers modulo ``6`` or ``9`` and therefore has zero
divisors and non-invertible elements::

  >>> K = GF(6)
  >>> K(3) * K(2) #doctest: +SKIP
  0 mod 6

It would be good to have a proper implementation of prime-power order finite
fields but this is not yet available in SymPy (contributions welcome!).

.. _Finite fields: https://en.wikipedia.org/wiki/Finite_field

Real and complex fields
=======================

The fields :ref:`RR` and :ref:`CC` are intended mathematically to correspond
to the `reals`_ and the `complex numbers`_, `\mathbb{R}` and `\mathbb{C}`
respectively. The implementation of these uses floating point arithmetic. In
practice this means that these are the domains that are used to represent
expressions containing floats. Elements of :ref:`RR` are instances of the
class :py:class:`~.RealElement` and have an ``mpf`` tuple which is used to
represent a float in ``mpmath``. Elements of :ref:`CC` are instances of
:py:class:`~.ComplexElement` and have an ``mpc`` tuple which is a pair of
``mpf`` tuples representing the real and imaginary parts. See the
`mpmath docs`_ for more about how floating point numbers are represented::

  >>> from sympy import RR, CC
  >>> xr = RR(3)
  >>> xr
  3.0
  >>> xr._mpf_
  (0, 3, 0, 2)
  >>> zc = CC(3+1j)
  >>> zc
  (3.0 + 1.0j)
  >>> zc._mpc_
  ((0, 3, 0, 2), (0, 1, 0, 1))

The use of approximate floating point arithmetic in these domains comes with
all of the usual pitfalls. Many algorithms in the :mod:`sympy.polys` module are
fundamentally designed for exact arithmetic making the use of these domains
potentially problematic::

  >>> RR('0.1') + RR('0.2') == RR('0.3')
  False

Since these are implemented using ``mpmath`` which is a multiprecision library
it is possible to create different domains with different working precisions.
The default domains :ref:`RR` and :ref:`CC` use 53 binary digits of precision
much like standard `double precision`_ floating point which corresponds to
approximately 15 decimal digits::

  >>> from sympy.polys.domains.realfield import RealField
  >>> RR.precision
  53
  >>> RR.dps
  15
  >>> RR(1) / RR(3)
  0.333333333333333
  >>> RR100 = RealField(100)
  >>> RR100.precision
  100
  >>> RR100.dps
  29
  >>> RR100(1) / RR100(3)
  0.33333333333333333333333333333

There is however a bug in the implementation of this so that actually a global
precision setting is used by all :py:class:`~.RealElement`. This means that
just creating ``RR100`` above has altered the global precision and we will
need to restore it in the doctest here::

  >>> RR(1) / RR(3)  # wrong result!
  0.33333333333333333333333333333
  >>> dummy = RealField(53)  # hack to restore precision
  >>> RR(1) / RR(3)  # restored
  0.333333333333333

(Obviously that should be fixed!)

.. _reals: https://en.wikipedia.org/wiki/Real_number
.. _complex numbers: https://en.wikipedia.org/wiki/Complex_number
.. _mpmath docs: https://mpmath.org/doc/current/technical.html#representation-of-numbers
.. _double precision: https://en.wikipedia.org/wiki/Double-precision_floating-point_format

Algebraic number fields
=======================

An `algebraic extension`_ of the rationals `\mathbb{Q}` is known as an
`algebraic number field`_ and these are implemented in sympy as :ref:`QQ(a)`.
The natural syntax for these would be something like ``QQ(sqrt(2))`` however
``QQ()`` is already overloaded as the constructor for elements of :ref:`QQ`.
These domains are instead created using the
:py:meth:`~.Domain.algebraic_field` method e.g.
``QQ.algebraic_field(sqrt(2))``. The resulting domain will be an instance of
:py:class:`~.AlgebraicField` with elements that are instances of
:py:class:`~.ANP`.

The printing support for these is less developed but we can use
:py:meth:`~.Domain.to_sympy` to take advantage of the corresponding
:py:class:`~.Expr` printing support::

  >>> K = QQ.algebraic_field(sqrt(2))
  >>> K
  QQ<sqrt(2)>
  >>> b = K.one + K.from_sympy(sqrt(2))
  >>> b  # doctest: +SKIP
  ANP([1, 1], [1, 0, -2], QQ)
  >>> K.to_sympy(b)
  1 + sqrt(2)
  >>> b ** 2  # doctest: +SKIP
  ANP([2, 3], [1, 0, -2], QQ)
  >>> K.to_sympy(b**2)
  2*sqrt(2) + 3

The raw printed display immediately shows the internal representation of the
elements as :py:class:`~.ANP` instances. The field `\mathbb{Q}(\sqrt{2})`
consists of numbers of the form `a\sqrt{2}+b` where `a` and `b` are rational
numbers. Consequently every number in this field can be represented as a pair
``(a, b)`` of elements of :ref:`QQ`. The domain element stores these two in a
list and also stores a list representation of the *minimal polynomial* for the
extension element `\sqrt{2}`. There is a sympy function :py:func:`~.minpoly`
that can compute the minimal polynomial of any algebraic expression over the
rationals::

  >>> from sympy import minpoly, Symbol
  >>> x = Symbol('x')
  >>> minpoly(sqrt(2), x)
  x**2 - 2

In the dense polynomial representation as a list of coefficients this
polynomial is represented as ``[1, 0, -2]`` as seen in the :py:class:`~.ANP`
display for the elements of ``QQ<sqrt(2)>`` above.

It is also possible to create an algebraic number field with multiple
generators such as `\mathbb{Q}(\sqrt{2},\sqrt{3})`::

  >>> K = QQ.algebraic_field(sqrt(2), sqrt(3))
  >>> K
  QQ<sqrt(2) + sqrt(3)>
  >>> sqrt2 = K.from_sympy(sqrt(2))
  >>> sqrt3 = K.from_sympy(sqrt(3))
  >>> p = (K.one + sqrt2) * (K.one + sqrt3)
  >>> p  # doctest: +SKIP
  ANP([1/2, 1, -3/2], [1, 0, -10, 0, 1], QQ)
  >>> K.to_sympy(p)
  1 + sqrt(2) + sqrt(3) + sqrt(6)
  >>> K.to_sympy(p**2)
  4*sqrt(6) + 6*sqrt(3) + 8*sqrt(2) + 12

Here the algebraic extension `\mathbb{Q}(\sqrt{2},\sqrt{3})` is converted to
the (isomorphic) `\mathbb{Q}(\sqrt{2}+\sqrt{3})` with a single generator
`\sqrt{2}+\sqrt{3}`. It is always possible to find a single generator like
this due to the `primitive element theorem`_. There is a sympy function
:py:func:`~.primitive_element` that can compute the minimal polynomial for a
primitive element of an extension::

  >>> from sympy import primitive_element, minpoly
  >>> e = primitive_element([sqrt(2), sqrt(3)], x)
  >>> e[0]
  x**4 - 10*x**2 + 1
  >>> e[0].subs(x, sqrt(2) + sqrt(3)).expand()
  0

The minimal polynomial ``x**4 - 10*x**2 + 1`` has the dense list representation
``[1, 0, -10, 0, 1]`` as seen in the :py:class:`~.ANP` output above. What the
primitive element theorem means is that all algebraic number fields can be
represented as an extension of the rationals by a single generator with some
minimal polynomial. Calculations over the algebraic number field only need to
take advantage of the minimal polynomial and that makes it possible to compute
all arithmetic operations and also to carry out higher level operations like
factorisation of polynomials.

.. _algebraic extension: https://en.wikipedia.org/wiki/Algebraic_extension
.. _algebraic number field: https://en.wikipedia.org/wiki/Algebraic_number_field
.. _primitive element theorem: https://en.wikipedia.org/wiki/Primitive_element_theorem

Polynomial ring domains
=======================

There are also domains implemented to represent a polynomial ring like
:ref:`K[x]` which is the domain of polynomials in the generator ``x`` with
coefficients over another domain ``K``::

  >>> from sympy import ZZ, symbols
  >>> x = symbols('x')
  >>> K = ZZ[x]
  >>> K
  ZZ[x]
  >>> x_dom = K(x)
  >>> x_dom + K.one
  x + 1

All the operations discussed before will work with elements of a polynomial
ring::

  >>> p = x_dom + K.one
  >>> p
  x + 1
  >>> p + p
  2*x + 2
  >>> p - p
  0
  >>> p * p
  x**2 + 2*x + 1
  >>> p ** 3
  x**3 + 3*x**2 + 3*x + 1
  >>> K.exquo(x_dom**2 - K.one, x_dom - K.one)
  x + 1

The internal representation of elements of ``K[x]`` is different from the way
that ordinary sympy (:py:class:`~.Expr`) expressions are represented. The
:py:class:`~.Expr` representation of any expression is as a tree e.g.::

  >>> from sympy import srepr
  >>> K = ZZ[x]
  >>> p_expr = x**2 + 2*x + 1
  >>> p_expr
  x**2 + 2*x + 1
  >>> srepr(p_expr)
  "Add(Pow(Symbol('x'), Integer(2)), Mul(Integer(2), Symbol('x')), Integer(1))"

Here the expression is a tree where the top node is an :py:class:`~.Add` and
its children nodes are :py:class:`~.Pow` etc. This tree representation makes
it possible to represent equivalent expressions in different ways e.g.::

  >>> x = symbols('x')
  >>> p_expr = x*(x + 1) + x
  >>> p_expr
  x*(x + 1) + x
  >>> p_expr.expand()
  x**2 + 2*x

By contrast the domain ``ZZ[x]`` represents only polynomials and does so by
simply storing the non-zero coefficients of the expanded polynomial (the
"sparse" polynomial representation). In particular elements of ``ZZ[x]`` are
represented as a Python ``dict``. Their type is :py:class:`~.PolyElement`
which is a subclass of ``dict``. Converting to a normal dict shows the
internal representation::

  >>> x = symbols('x')
  >>> K = ZZ[x]
  >>> x_dom = K(x)
  >>> p_dom = K(3)*x_dom**2 + K(2)*x_dom + K(7)
  >>> p_dom
  3*x**2 + 2*x + 7
  >>> dict(p_dom)
  {(0,): 7, (1,): 2, (2,): 3}

This internal form makes it impossible to represent unexpanded multiplications
so any multiplication of elements of ``ZZ[x]`` will always be
expanded::

  >>> x = symbols('x')
  >>> K = ZZ[x]
  >>> x_dom = K(x)
  >>> p_expr = x * (x + 1) + x
  >>> p_expr
  x*(x + 1) + x
  >>> p_dom = x_dom * (x_dom + K.one) + x_dom
  >>> p_dom
  x**2 + 2*x

These same considerations apply to powers::

  >>> (x + 1) ** 2
  (x + 1)**2
  >>> (x_dom + K.one) ** 2
  x**2 + 2*x + 1

We can also construct multivariate polynomial rings::

  >>> x, y = symbols('x, y')
  >>> K = ZZ[x,y]
  >>> xk = K(x)
  >>> yk = K(y)
  >>> xk**2*yk + xk + yk
  x**2*y + x + y

It is also possible to construct nested polynomial rings (although it is less
efficient). The ring ``K[x][y]`` is formally equivalent to ``K[x,y]`` although
their implementations in sympy are different::

  >>> K = ZZ[x][y]
  >>> p = K(x**2 + x*y + y**2)
  >>> p
  y**2 + x*y + x**2
  >>> dict(p)
  {(0,): x**2, (1,): x, (2,): 1}

Here the coefficients like ``x**2`` are instances of :py:class:`~.PolyElement`
as well so this is a ``dict`` where the values are also dicts. The full
representation is more like::

  >>> {k: dict(v) for k, v in p.items()}
  {(0,): {(2,): 1}, (1,): {(1,): 1}, (2,): {(0,): 1}}

The multivariate ring domain ``ZZ[x,y]`` has a more efficient representation
as a single flattened ``dict``::

  >>> K = ZZ[x,y]
  >>> p = K(x**2 + x*y + y**2)
  >>> p
  x**2 + x*y + y**2
  >>> dict(p)
  {(0, 2): 1, (1, 1): 1, (2, 0): 1}

The difference in efficiency between these representations grows as the number
of generators increases i.e. ``ZZ[x,y,z,t,...]`` vs ``ZZ[x][y][z][t]...``.

Old (dense) polynomial rings
============================

In the last section we saw that the domain representation of a polynomial ring
like :ref:`K[x]` uses a sparse representation of a polynomial as a dict
mapping monomial exponents to coefficients. There is also an older version of
:ref:`K[x]` that uses the dense :ref:`dmp-representation`. We can create these
two versions of :ref:`K[x]` using :py:meth:`~.Domain.poly_ring` and
:py:meth:`~.Domain.old_poly_ring` where the syntax ``K[x]`` is equivalent to
``K.poly_ring(x)``::

  >>> K1 = ZZ.poly_ring(x)
  >>> K2 = ZZ.old_poly_ring(x)
  >>> K1
  ZZ[x]
  >>> K2
  ZZ[x]
  >>> K1 == ZZ[x]
  True
  >>> K2 == ZZ[x]
  False
  >>> p1 = K1.from_sympy(x**2 + 1)
  >>> p2 = K2.from_sympy(x**2 + 1)
  >>> p1
  x**2 + 1
  >>> p2  # doctest: +SKIP
  DMP_Python([1, 0, 1], ZZ)
  >>> type(K1)
  <class 'sympy.polys.domains.polynomialring.PolynomialRing'>
  >>> type(p1)
  <class 'sympy.polys.rings.PolyElement'>
  >>> type(K2)
  <class 'sympy.polys.domains.old_polynomialring.GlobalPolynomialRing'>
  >>> type(p2)  # doctest: +SKIP
  <class 'sympy.polys.polyclasses.DMP_Python'>

The internal representation of the old polynomial ring domain is the
:py:class:`~.DMP` representation as a list of (lists of) coefficients::

  >>> repr(p2)  # doctest: +SKIP
  'DMP_Python([1, 0, 1], ZZ, ZZ[x])'

The most notable use of the :py:class:`~.DMP` representation of polynomials is
as the internal representation used by :py:class:`~.Poly` (this is discussed
later in this page of the docs).

PolyRing vs PolynomialRing
==========================

You might just want to perform calculations in some particular polynomial ring
without being concerned with implementing something that works for arbitrary
domains. In that case you can construct the ring more directly with the
:py:func:`~.ring` function::

  >>> from sympy import ring
  >>> K, xr, yr = ring([x, y], ZZ)
  >>> K
  Polynomial ring in x, y over ZZ with lex order
  >>> xr**2 - yr**2
  x**2 - y**2
  >>> (xr**2 - yr**2) // (xr - yr)
  x + y

The object ``K`` here represents the ring and is an instance of
:py:class:`~.PolyRing` but is not a **polys domain** (it is not an instance of
a subclass of :py:class:`~.Domain` so it can not be used with
:py:class:`~.Poly`). In this way the implementation of polynomial rings that
is used in the domain system can be used independently of the domain system.

The purpose of the domain system is to provide a unified interface for working
with and converting between different representations of expressions. To make
the :py:class:`~.PolyRing` implementation usable in that context the
:py:class:`~.PolynomialRing` class is a wrapper around the
:py:class:`~.PolyRing` class that provides the interface expected in the
domain system. That makes this implementation of polynomial rings usable as
part of the broader codebase that is designed to work with expressions from
different domains. The domain for polynomial rings is a distinct object from
the ring returned by :py:func:`~.ring` although both have the same elements::

  >>> K, xr, yr = ring([x, y], ZZ)
  >>> K
  Polynomial ring in x, y over ZZ with lex order
  >>> K2 = ZZ[x,y]
  >>> K2
  ZZ[x,y]
  >>> K2.ring
  Polynomial ring in x, y over ZZ with lex order
  >>> K2.ring == K
  True
  >>> K(x+y)
  x + y
  >>> K2(x+y)
  x + y
  >>> type(K(x+y))
  <class 'sympy.polys.rings.PolyElement'>
  >>> type(K2(x+y))
  <class 'sympy.polys.rings.PolyElement'>
  >>> K(x+y) == K2(x+y)
  True

Rational function fields
========================

Some domains are classified as fields and others are not. The principal
difference between a field and a non-field domain is that in a field it is
always possible to divide any element by any nonzero element. It is usually
possible to convert any domain to a field that contains that domain with the
:py:meth:`~.Domain.get_field` method::

  >>> from sympy import ZZ, QQ, symbols
  >>> x, y = symbols('x, y')
  >>> ZZ.is_Field
  False
  >>> QQ.is_Field
  True
  >>> QQ[x]
  QQ[x]
  >>> QQ[x].is_Field
  False
  >>> QQ[x].get_field()
  QQ(x)
  >>> QQ[x].get_field().is_Field
  True
  >>> QQ.frac_field(x)
  QQ(x)

This introduces a new kind of domain :ref:`K(x)` representing a rational
function field in the generator ``x`` over another domain ``K``. It is not
possible to construct the domain ``QQ(x)`` with the ``()`` syntax so the
easiest ways to create it are using the domain methods
:py:meth:`~.Domain.frac_field` (``QQ.frac_field(x)``) or
:py:meth:`~.Domain.get_field` (``QQ[x].get_field()``). The
:py:meth:`~.Domain.frac_field` method is the more direct approach.

The rational function field :ref:`K(x)` is an instance of
:py:class:`~.RationalField`. This domain represents functions of the form
`p(x) / q(x)` for polynomials `p` and `q`. The domain elements are
represented as a pair of polynomials in :ref:`K[x]`::

  >>> K = QQ.frac_field(x)
  >>> xk = K(x)
  >>> f = xk / (K.one + xk**2)
  >>> f
  x/(x**2 + 1)
  >>> f.numer
  x
  >>> f.denom
  x**2 + 1
  >>> QQ[x].of_type(f.numer)
  True
  >>> QQ[x].of_type(f.denom)
  True

Cancellation between the numerator and denominator is automatic in this
field::

  >>> p1 = xk**2 - 1
  >>> p2 = xk - 1
  >>> p1
  x**2 - 1
  >>> p2
  x - 1
  >>> p1 / p2
  x + 1

Computing this cancellation can be slow which makes rational function fields
potentially slower than polynomial rings or algebraic fields.

Just like in the case of polynomial rings there is both a new (sparse) and old
(dense) version of fraction fields::

  >>> K1 = QQ.frac_field(x)
  >>> K2 = QQ.old_frac_field(x)
  >>> K1
  QQ(x)
  >>> K2
  QQ(x)
  >>> type(K1)
  <class 'sympy.polys.domains.fractionfield.FractionField'>
  >>> type(K2)
  <class 'sympy.polys.domains.old_fractionfield.FractionField'>

Also just like in the case of polynomials rings the implementation of rational
function fields can be used independently of the domain system::

  >>> from sympy import field
  >>> K, xf, yf = field([x, y], ZZ)
  >>> xf / (1 - yf)
  -x/(y - 1)

Here ``K`` is an instance of :py:class:`~.FracField` rather than
:py:class:`~.RationalField` as it would be for the domain ``ZZ(x,y)``.

Expression domain
=================

The final domain to consider is the "expression domain" which is known as
:ref:`EX`. Expressions that can not be represented using the other domains can
be always represented using the expression domain. An element of :ref:`EX` is
actually just a wrapper around a :py:class:`~.Expr` instance::

  >>> from sympy import EX
  >>> p = EX.from_sympy(1 + x)
  >>> p
  EX(x + 1)
  >>> type(p)
  <class 'sympy.polys.domains.expressiondomain.ExpressionDomain.Expression'>
  >>> p.ex
  x + 1
  >>> type(p.ex)
  <class 'sympy.core.add.Add'>

For other domains the domain representation of expressions is usually more
efficient than the tree representation used by :py:class:`~.Expr`. In
:ref:`EX` the internal representation is :py:class:`~.Expr` so it is clearly
not more efficient. The purpose of the :ref:`EX` domain is to be able to wrap
up arbitrary expressions in an interface that is consistent with the other
domains. The :ref:`EX` domain is used as a fallback when an appropriate domain
can not be found. Although this does not offer any particular efficiency it
does allow the algorithms that are implemented to work over arbitrary domains
to be usable when working with expressions that do not have an appropriate
domain representation.

Choosing a domain
=================

In the workflow described above the idea is to start with some sympy
expressions, choose a domain and convert all the expressions into that domain
in order to perform some calculation. The obvious question that arises is how
to choose an appropriate domain to represent some sympy expressions. For this
there is a function :py:func:`~.construct_domain` which takes a list of
expressions and will choose a domain and convert all of the expressions to
that domain::

  >>> from sympy import construct_domain, Integer
  >>> elements_sympy = [Integer(3), Integer(2)]  # elements as Expr instances
  >>> elements_sympy
  [3, 2]
  >>> K, elements_K = construct_domain(elements_sympy)
  >>> K
  ZZ
  >>> elements_K
  [3, 2]
  >>> type(elements_sympy[0])
  <class 'sympy.core.numbers.Integer'>
  >>> type(elements_K[0])  # doctest: +SKIP
  <class 'int'>

In this example we see that the two integers ``3`` and ``2`` can be
represented in the domain :ref:`ZZ`. The expressions have been converted to
elements of that domain which in this case means the ``int`` type rather than
instances of :py:class:`~.Expr`. It is not necessary to explicitly create
:py:class:`~.Expr` instances when the inputs can be sympified so e.g.
``construct_domain([3, 2])`` would give the same output as above.

Given more complicated inputs :py:func:`~.construct_domain` will choose more
complicated domains::

  >>> from sympy import Rational, symbols
  >>> x, y = symbols('x, y')
  >>> construct_domain([Rational(1, 2), Integer(3)])[0]
  QQ
  >>> construct_domain([2*x, 3])[0]
  ZZ[x]
  >>> construct_domain([x/2, 3])[0]
  QQ[x]
  >>> construct_domain([2/x, 3])[0]
  ZZ(x)
  >>> construct_domain([x, y])[0]
  ZZ[x,y]

If any noninteger rational numbers are found in the inputs then the ground
domain will be :ref:`QQ` rather than :ref:`ZZ`. If any symbol is found in the
inputs then a :py:class:`~.PolynomialRing` will be created. A multivariate
polynomial ring such as ``QQ[x,y]`` can also be created if there are multiple
symbols in the inputs. If any symbols appear in the denominators then a
:py:class:`~.RationalField` like ``QQ(x)`` will be created instead.

Some of the domains above are fields and others are (non-field) rings. In some
contexts it is necessary to have a field domain so that division is possible
and for this :py:func:`~.construct_domain` has an option ``field=True`` which
will force the construction of a field domain even if the expressions can all
be represented in a non-field ring::

  >>> construct_domain([1, 2], field=True)[0]
  QQ
  >>> construct_domain([2*x, 3], field=True)[0]
  ZZ(x)
  >>> construct_domain([x/2, 3], field=True)[0]
  ZZ(x)
  >>> construct_domain([2/x, 3], field=True)[0]
  ZZ(x)
  >>> construct_domain([x, y], field=True)[0]
  ZZ(x,y)

By default :py:func:`~.construct_domain` will not construct an algebraic
extension field and will instead use the :ref:`EX` domain
(:py:class:`~.ExpressionDomain`). The keyword argument ``extension=True`` can
be used to construct an :py:class:`~.AlgebraicField` if the inputs are
irrational but algebraic::

  >>> from sympy import sqrt
  >>> construct_domain([sqrt(2)])[0]
  EX
  >>> construct_domain([sqrt(2)], extension=True)[0]
  QQ<sqrt(2)>
  >>> construct_domain([sqrt(2), sqrt(3)], extension=True)[0]
  QQ<sqrt(2) + sqrt(3)>

When there are algebraically independent transcendentals in the inputs a
:py:class:`~.PolynomialRing` or :py:class:`~.RationalField` will be
constructed treating those transcendentals as generators::

  >>> from sympy import sin, cos
  >>> construct_domain([sin(x), y])[0]
  ZZ[y,sin(x)]

However if there is a possibility that the inputs are not algebraically
independent then the domain will be :ref:`EX`::

  >>> construct_domain([sin(x), cos(x)])[0]
  EX

Here ``sin(x)`` and ``cos(x)`` are not algebraically independent since
``sin(x)**2 + cos(x)**2 = 1``.

Converting elements between different domains
=============================================

It is often useful to combine calculations performed over different domains.
However just as it is important to avoid mixing domain elements with normal
sympy expressions and other Python types it is also important to avoid mixing
elements from different domains. The :py:meth:`~.Domain.convert_from` method
is used to convert elements from one domain into elements of another domain::

  >>> num_zz = ZZ(3)
  >>> ZZ.of_type(num_zz)
  True
  >>> num_qq = QQ.convert_from(num_zz, ZZ)
  >>> ZZ.of_type(num_qq)
  False
  >>> QQ.of_type(num_qq)
  True

The :py:meth:`~.Domain.convert` method can be called without specifying the
source domain as the second argument e.g.::

  >>> QQ.convert(ZZ(2))
  2

This works because :py:meth:`~.Domain.convert` can check the type of ``ZZ(2)``
and can try to work out what domain (:ref:`ZZ`) it is an element of. Certain
domains like :ref:`ZZ` and :ref:`QQ` are treated as special cases to make this work.
Elements of more complicated domains are instances of subclasses of
:py:class:`~.DomainElement` which has a :py:meth:`~.DomainElement.parent`
method that can identify the domain that the element belongs to. For example
in the polynomial ring ``ZZ[x]`` we have::

  >>> from sympy import ZZ, Symbol
  >>> x = Symbol('x')
  >>> K = ZZ[x]
  >>> K
  ZZ[x]
  >>> p = K(x) + K.one
  >>> p
  x + 1
  >>> type(p)
  <class 'sympy.polys.rings.PolyElement'>
  >>> p.parent()
  ZZ[x]
  >>> p.parent() == K
  True

It is more efficient though to call :py:meth:`~.Domain.convert_from` with the
source domain specified as the second argument::

  >>> QQ.convert_from(ZZ(2), ZZ)
  2

Unifying domains
================

When we want to combine elements from two different domains and perform mixed
calculations with them we need to

#. Choose a new domain that can represent all elements of both.
#. Convert all elements to the new domain.
#. Perform the calculation in the new domain.

The key question arising from point 1. is how to choose a domain that can
represent the elements of both domains. For this there is the
:py:meth:`~.Domain.unify` method::

  >>> x1, K1 = ZZ(2), ZZ
  >>> y2, K2 = QQ(3, 2), QQ
  >>> K1
  ZZ
  >>> K2
  QQ
  >>> K3 = K1.unify(K2)
  >>> K3
  QQ
  >>> x3 = K3.convert_from(x1, K1)
  >>> y3 = K3.convert_from(y2, K2)
  >>> x3 + y3
  7/2

The :py:meth:`~.Domain.unify` method will find a domain that encompasses both
domains so in this example ``ZZ.unify(QQ)`` gives :ref:`QQ` because every element
of :ref:`ZZ` can be represented as an element of :ref:`QQ`. This means that all
inputs (``x1`` and ``y2``) can be converted to the elements of the common
domain ``K3`` (as ``x3`` and ``y3``). Once in the common domain we can safely
use arithmetic operations like ``+``. In this example one domain is a superset
of the other and we see that ``K1.unify(K2) == K2`` so it is not actually
necessary to convert ``y2``. In general though ``K1.unify(K2)`` can give a new
domain that is not equal to either ``K1`` or ``K2``.

The :py:meth:`~.Domain.unify` method understands how to combine different
polynomial ring domains and how to unify the base domain::

  >>> ZZ[x].unify(ZZ[y])
  ZZ[x,y]
  >>> ZZ[x,y].unify(ZZ[y])
  ZZ[x,y]
  >>> ZZ[x].unify(QQ)
  QQ[x]

It is also possible to unify algebraic fields and rational function fields as
well::

  >>> K1 = QQ.algebraic_field(sqrt(2))[x]
  >>> K2 = QQ.algebraic_field(sqrt(3))[y]
  >>> K1
  QQ<sqrt(2)>[x]
  >>> K2
  QQ<sqrt(3)>[y]
  >>> K1.unify(K2)
  QQ<sqrt(2) + sqrt(3)>[x,y]
  >>> QQ.frac_field(x).unify(ZZ[y])
  ZZ(x,y)

Internals of a Poly
===================

We are now in a position to understand how the :py:class:`~.Poly` class works
internally. This is the public interface of :py:class:`~.Poly`::

  >>> from sympy import Poly, symbols, ZZ
  >>> x, y, z, t = symbols('x, y, z, t')
  >>> p = Poly(x**2 + 1, x, domain=ZZ)
  >>> p
  Poly(x**2 + 1, x, domain='ZZ')
  >>> p.gens
  (x,)
  >>> p.domain
  ZZ
  >>> p.all_coeffs()
  [1, 0, 1]
  >>> p.as_expr()
  x**2 + 1

This is the internal implementation of :py:class:`~.Poly`::

  >>> d = p.rep  # internal representation of Poly
  >>> d  # doctest: +SKIP
  DMP_Python([1, 0, 1], ZZ)
  >>> d.rep      # internal representation of DMP  # doctest: +SKIP
  [1, 0, 1]
  >>> type(d.rep)  # doctest: +SKIP
  <class 'list'>
  >>> type(d.rep[0])  # doctest: +SKIP
  <class 'int'>
  >>> d.dom
  ZZ

The internal representation of a :py:class:`~.Poly` instance is an instance of
:py:class:`~.DMP` which is the class used for domain elements in the old
polynomial ring domain :py:meth:`~.Domain.old_poly_ring`. This represents the
polynomial as a list of coefficients which are themselves elements of a domain
and keeps a reference to their domain (:ref:`ZZ` in this example).

Choosing a domain for a Poly
============================

If the domain is not specified for the :py:class:`~.Poly` constructor then it
is inferred using :py:func:`~.construct_domain`. Arguments like ``field=True``
are passed along to :py:func:`~.construct_domain`::

  >>> from sympy import sqrt
  >>> Poly(x**2 + 1, x)
  Poly(x**2 + 1, x, domain='ZZ')
  >>> Poly(x**2 + 1, x, field=True)
  Poly(x**2 + 1, x, domain='QQ')
  >>> Poly(x**2/2 + 1, x)
  Poly(1/2*x**2 + 1, x, domain='QQ')
  >>> Poly(x**2 + sqrt(2), x)
  Poly(x**2 + sqrt(2), x, domain='EX')
  >>> Poly(x**2 + sqrt(2), x, extension=True)
  Poly(x**2 + sqrt(2), x, domain='QQ<sqrt(2)>')

It is also possible to use the extension argument to specify generators of an
extension even if no extension is required to represent the coefficients
although this does not work when using :py:func:`~.construct_domain` directly.
A list of extension elements will be passed to :py:func:`~.primitive_element`
to create an appropriate :py:class:`~.AlgebraicField` domain::

  >>> from sympy import construct_domain
  >>> Poly(x**2 + 1, x)
  Poly(x**2 + 1, x, domain='ZZ')
  >>> Poly(x**2 + 1, x, extension=sqrt(2))
  Poly(x**2 + 1, x, domain='QQ<sqrt(2)>')
  >>> Poly(x**2 + 1, x, extension=[sqrt(2), sqrt(3)])
  Poly(x**2 + 1, x, domain='QQ<sqrt(2) + sqrt(3)>')
  >>> construct_domain([1, 0, 1], extension=sqrt(2))[0]
  ZZ

(Perhaps :py:func:`~.construct_domain` should do the same as
:py:class:`~.Poly` here...)

Choosing generators
===================

If there are symbols other than the generators then a polynomial ring or
rational function field domain will be created. The domain used for the
coefficients in this case is the sparse ("new") polynomial ring::

  >>> p = Poly(x**2*y + z, x)
  >>> p
  Poly(y*x**2 + z, x, domain='ZZ[y,z]')
  >>> p.gens
  (x,)
  >>> p.domain
  ZZ[y,z]
  >>> p.domain == ZZ[y,z]
  True
  >>> p.domain == ZZ.poly_ring(y, z)
  True
  >>> p.domain == ZZ.old_poly_ring(y, z)
  False
  >>> p.rep.rep  # doctest: +SKIP
  [y, 0, z]
  >>> p.rep.rep[0]  # doctest: +SKIP
  y
  >>> type(p.rep.rep[0])  # doctest: +SKIP
  <class 'sympy.polys.rings.PolyElement'>
  >>> dict(p.rep.rep[0])  # doctest: +SKIP
  {(1, 0): 1}

What we have here is a strange hybrid of dense and sparse implementations. The
:py:class:`~.Poly` instance considers itself to be an univariate polynomial in
the generator ``x`` but with coefficients from the domain ``ZZ[y,z]``. The
internal representation of the :py:class:`~.Poly` is a list of coefficients in
the "dense univariate polynomial" (DUP) format. However each coefficient is
implemented as a sparse polynomial in ``y`` and ``z``.

If we make ``x``, ``y`` and ``z`` all be generators for the :py:class:`~.Poly`
then we get a fully dense DMP list of lists of lists representation::

  >>> p = Poly(x**2*y + z, x, y, z)
  >>> p
  Poly(x**2*y + z, x, y, z, domain='ZZ')
  >>> p.rep
  DMP_Python([[[1], []], [[]], [[1, 0]]], ZZ)
  >>> p.rep.rep  # doctest: +SKIP
  [[[1], []], [[]], [[1, 0]]]
  >>> p.rep.rep[0][0][0]  # doctest: +SKIP
  1
  >>> type(p.rep.rep[0][0][0])  # doctest: +SKIP
  <class 'int'>

On the other hand we can make a :py:class:`~.Poly` with a fully sparse
representation by choosing a generator that is not in the expression at all::

  >>> p = Poly(x**2*y + z, t)
  >>> p
  Poly(x**2*y + z, t, domain='ZZ[x,y,z]')
  >>> p.rep
  DMP_Python([x**2*y + z], ZZ[x,y,z])
  >>> p.rep.rep[0]  # doctest: +SKIP
  x**2*y + z
  >>> type(p.rep.rep[0])  # doctest: +SKIP
  <class 'sympy.polys.rings.PolyElement'>
  >>> dict(p.rep.rep[0])  # doctest: +SKIP
  {(0, 0, 1): 1, (2, 1, 0): 1}

If no generators are provided to the :py:class:`~.Poly` constructor then it
will attempt to choose generators so that the expression is polynomial in
those. In the common case that the expression is a polynomial expression in
some symbols then those symbols will be taken as generators. However other
non-symbol expressions can also be taken as generators::

  >>> Poly(x**2*y + z)
  Poly(x**2*y + z, x, y, z, domain='ZZ')
  >>> from sympy import pi, exp
  >>> Poly(exp(x) + exp(2*x) + 1)
  Poly((exp(x))**2 + (exp(x)) + 1, exp(x), domain='ZZ')
  >>> Poly(pi*x)
  Poly(x*pi, x, pi, domain='ZZ')
  >>> Poly(pi*x, x)
  Poly(pi*x, x, domain='ZZ[pi]')

Algebraically dependent generators
==================================

Taking ``exp(x)`` or ``pi`` as generators for a :py:class:`~.Poly` or for its
polynomial ring domain is mathematically valid because these objects are
transcendental and so the ring extension containing them is isomorphic to a
polynomial ring. Since ``x`` and ``exp(x)`` are algebraically independent it
is also valid to use both as generators for the same :py:class:`~.Poly`.
However some other combinations of generators are invalid such as ``x`` and
``sqrt(x)`` or ``sin(x)`` and ``cos(x)``. These examples are invalid  because
the generators are not algebraically independent (e.g. ``sqrt(x)**2 = x`` and
``sin(x)**2 + cos(x)**2 = 1``). The implementation is not able to detect these
algebraic relationships though::

  >>> from sympy import sin, cos, sqrt
  >>> Poly(x*exp(x))      # fine
  Poly(x*(exp(x)), x, exp(x), domain='ZZ')
  >>> Poly(sin(x)+cos(x)) # not fine
  Poly((cos(x)) + (sin(x)), cos(x), sin(x), domain='ZZ')
  >>> Poly(x + sqrt(x))   # not fine
  Poly(x + (sqrt(x)), x, sqrt(x), domain='ZZ')

Calculations with a :py:class:`~.Poly` such as this are unreliable because
zero-testing will not work properly in this implementation::

  >>> p1 = Poly(x, x, sqrt(x))
  >>> p2 = Poly(sqrt(x), x, sqrt(x))
  >>> p1
  Poly(x, x, sqrt(x), domain='ZZ')
  >>> p2
  Poly((sqrt(x)), x, sqrt(x), domain='ZZ')
  >>> p3 = p1 - p2**2
  >>> p3                  # should be zero...
  Poly(x - (sqrt(x))**2, x, sqrt(x), domain='ZZ')
  >>> p3.as_expr()
  0

This aspect of :py:class:`~.Poly` could be improved by:

#. Expanding the domain system with new domains that can represent more
   classes of algebraic extension.
#. Improving the detection of algebraic dependencies in
   :py:func:`~.construct_domain`.
#. Improving the automatic selection of generators.

Examples of the above are that it would be useful to have a domain that can
represent more general algebraic extensions (:py:class:`~.AlgebraicField` is
only for extensions of :ref:`QQ`). Improving the detection of algebraic
dependencies is harder but at least common cases like ``sin(x)`` and
``cos(x)`` could be handled. When choosing generators it should be possible to
recognise that ``sqrt(x)`` can be the only generator for ``x + sqrt(x)``::

  >>> Poly(x + sqrt(x))            # this could be improved!
  Poly(x + (sqrt(x)), x, sqrt(x), domain='ZZ')
  >>> Poly(x + sqrt(x), sqrt(x))   # this could be improved!
  Poly((sqrt(x)) + x, sqrt(x), domain='ZZ[x]')
