.. _polysbasics:
=================================
Basic functionality of the module
=================================
Introduction
============
This tutorial tries to give an overview of the functionality concerning
polynomials within SymPy. All code examples assume::
>>> from sympy import *
>>> x, y, z = symbols('x,y,z')
>>> import sys
>>> sys.displayhook = pprint
Basic functionality
===================
These functions provide different algorithms dealing with polynomials in the
form of SymPy expression, like symbols, sums etc.
Division

The function :func:`div` provides division of polynomials with remainder.
That is, for polynomials ``f`` and ``g``, it computes ``q`` and ``r``, such
that $f = g \cdot q + r$ and $\deg(r) < q$. For polynomials in one variables
with coefficients in a field, say, the rational numbers, ``q`` and ``r`` are
uniquely defined this way::
>>> f = 5*x**2 + 10*x + 3
>>> g = 2*x + 2
>>> q, r = div(f, g, domain='QQ')
>>> q
5*x 5
 + 
2 2
>>> r
2
>>> (q*g + r).expand()
2
5*x + 10*x + 3
As you can see, ``q`` has a noninteger coefficient. If you want to do division
only in the ring of polynomials with integer coefficients, you can specify an
additional parameter::
>>> q, r = div(f, g, domain='ZZ')
>>> q
0
>>> r
2
5*x + 10*x + 3
But be warned, that this ring is no longer Euclidean and that the degree of the
remainder doesn't need to be smaller than that of ``f``. Since 2 doesn't divide 5,
$2 x$ doesn't divide $5 x^2$, even if the degree is smaller. But::
>>> g = 5*x + 1
>>> q, r = div(f, g, domain='ZZ')
>>> q
x
>>> r
9*x + 3
>>> (q*g + r).expand()
2
5*x + 10*x + 3
This also works for polynomials with multiple variables::
>>> f = x*y + y*z
>>> g = 3*x + 3*z
>>> q, r = div(f, g, domain='QQ')
>>> q
y

3
>>> r
0
In the last examples, all of the three variables ``x``, ``y`` and ``z`` are
assumed to be variables of the polynomials. But if you have some unrelated
constant as coefficient, you can specify the variables explicitly::
>>> a, b, c = symbols('a,b,c')
>>> f = a*x**2 + b*x + c
>>> g = 3*x + 2
>>> q, r = div(f, g, domain='QQ')
>>> q
a*x 2*a b
   + 
3 9 3
>>> r
4*a 2*b
   + c
9 3
GCD and LCM

With division, there is also the computation of the greatest common divisor and
the least common multiple.
When the polynomials have integer coefficients, the contents' gcd is also
considered::
>>> f = 12*(x + 1)*x
>>> g = 16*x**2
>>> gcd(f, g)
4*x
It also works with multiple variables. In this case, the variables are ordered
alphabetically, be default, which has influence on the leading coefficient::
>>> f = x*y/2 + y**2
>>> g = 3*x + 6*y
>>> gcd(f, g)
x + 2*y
The lcm is connected with the gcd and one can be computed using the other::
>>> f = x*y**2 + x**2*y
>>> g = x**2*y**2
>>> gcd(f, g)
x*y
>>> lcm(f, g)
3 2 2 3
x *y + x *y
>>> (f*g).expand()
4 3 3 4
x *y + x *y
>>> (gcd(f, g, x, y)*lcm(f, g, x, y)).expand()
4 3 3 4
x *y + x *y
Squarefree factorization

The squarefree factorization of a univariate polynomial is the product of all
factors (not necessarily irreducible) of degree 1, 2 etc.::
>>> f = 2*x**2 + 5*x**3 + 4*x**4 + x**5
>>> sqf_list(f)
(1, [(x + 2, 1), (x, 2), (x + 1, 2)])
>>> sqf(f)
2 2
x *(x + 1) *(x + 2)
Factorization

This function provides factorization of univariate and multivariate polynomials
with rational coefficients::
>>> factor(x**4/2 + 5*x**3/12  x**2/3)
2
x *(2*x  1)*(3*x + 4)

12
>>> factor(x**2 + 4*x*y + 4*y**2)
2
(x + 2*y)
Groebner bases

Buchberger's algorithm is implemented, supporting various monomial orders::
>>> groebner([x**2 + 1, y**4*x + x**3], x, y, order='lex')
2 4
[x + 1, y  1]
>>> groebner([x**2 + 1, y**4*x + x**3, x*y*z**3], x, y, z, order='grevlex')
4 3 2
[y  1, z , x + 1]
Solving Equations

We have (incomplete) methods to find the complex or even symbolic roots of
polynomials and to solve some systems of polynomial equations::
>>> from sympy import roots, solve_poly_system
>>> solve(x**3 + 2*x + 3, x)
____ ____
1 \/ 11 *I 1 \/ 11 *I
[1,   ,  + ]
2 2 2 2
>>> p = Symbol('p')
>>> q = Symbol('q')
>>> sorted(solve(x**2 + p*x + q, x))
__________ __________
/ 2 / 2
p \/ p  4*q p \/ p  4*q
[  + ,    ]
2 2 2 2
>>> solve_poly_system([y  x, x  5], x, y)
[(5, 5)]
>>> solve_poly_system([y**2  x**3 + 1, y*x], x, y)
___ ___
1 \/ 3 *I 1 \/ 3 *I
[(0, I), (0, I), (1, 0), (  + , 0), (   , 0)]
2 2 2 2