Polynomials

We show here some functions, that provide different algorithms dealing with polynomials in the form of Diofant expression.

Note, that coefficients of a polynomial can be elements of any commutative ring, this ring is called the domain of the polynomial ring and may be specified as a keyword parameter for functions. Polynomial generators also can be specified via an arbitrary number of arguments after required arguments of functions.

Division

The function 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, g = 5*x**2 + 10*x + 3, 2*x + 2

>>> div(f, g)
⎛5⋅x   5    ⎞
⎜─── + ─, -2⎟
⎝ 2    2    ⎠
>>> expand(_[0]*g + _[1])
   2
5⋅x  + 10⋅x + 3

As you can see, q has a non-integer coefficient. If you want to do division only in the ring of polynomials with integer coefficients, you can specify an additional parameter:

>>> div(f, g, domain=ZZ)
⎛      2           ⎞
⎝0, 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

>>> div(f, g, domain=ZZ)
(x, 9⋅x + 3)
>>> expand(_[0]*g + _[1])
   2
5⋅x  + 10⋅x + 3

This also works for polynomials with multiple variables:

>>> div(x*y + y*z, 3*x + 3*z)
⎛y   ⎞
⎜─, 0⎟
⎝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:

>>> gcd((12*x + 12)*x, 16*x**2)
4⋅x

But if the polynomials have rational coefficients, then the returned polynomial is monic:

>>> gcd(3*x**2/2, 9*x/4)
x

Symbolic exponents are supported:

>>> gcd(2*x**(n + 4) - x**(n + 2), 4*x**(n + 1) + 3*x**n)
 n
x

It also works with multiple variables. In this case, the variables are ordered alphabetically, be default, which has influence on the leading coefficient:

>>> gcd(x*y/2 + y**2, 3*x + 6*y)
x + 2⋅y

The lcm is connected with the gcd and one can be computed using the other:

>>> f, g = x*y**2 + x**2*y, x**2*y**2

>>> gcd(f, g)
x⋅y
>>> lcm(f, g)
 3  2    2  3
x ⋅y  + x ⋅y
>>> expand(f*g)
 4  3    3  4
x ⋅y  + x ⋅y
>>> expand(gcd(f, g, x, y)*lcm(f, g, x, y))
 4  3    3  4
x ⋅y  + x ⋅y

Square-free factorization

The square-free factorization of a univariate polynomial is the product of all factors (not necessarily irreducible) of degree 1, 2 etc.:

>>> sqf(2*x**2 + 5*x**3 + 4*x**4 + x**5)
 2        2
x ⋅(x + 1) ⋅(x + 2)

Factorization

Factorization supported over different domains, lets compute one for the finite field with characteristic 5, rational field or its algebraic extension:

>>> f = x**4 - 3*x**2 + 1

>>> factor(f, modulus=5)
       2        2
(x - 2) ⋅(x + 2)
>>> factor(f)
⎛ 2        ⎞ ⎛ 2        ⎞
⎝x  - x - 1⎠⋅⎝x  + x - 1⎠
>>> factor(f, extension=GoldenRatio)
(x - φ)⋅(x + φ)⋅(x - 1 + φ)⋅(x - φ + 1)

You also may use gaussian keyword to obtain a factorization over Gaussian rationals:

>>> factor(4*x**4 + 8*x**3 + 77*x**2 + 18*x + 153, gaussian=True)
  ⎛    3⋅ⅈ⎞ ⎛    3⋅ⅈ⎞
4⋅⎜x - ───⎟⋅⎜x + ───⎟⋅(x + 1 - 4⋅ⅈ)⋅(x + 1 + 4⋅ⅈ)
  ⎝     2 ⎠ ⎝     2 ⎠

Computing with multivariate polynomials over various domains is as simple as in univariate case.

>>> factor(x**2 + 4*x*y + 4*y**2)
         2
(x + 2⋅y)
>>> factor(x**3 + y**3, extension=sqrt(-3))
        ⎛      ⎛        ___  ⎞⎞ ⎛      ⎛        ___  ⎞⎞
        ⎜      ⎜  1   ╲╱ 3 ⋅ⅈ⎟⎟ ⎜      ⎜  1   ╲╱ 3 ⋅ⅈ⎟⎟
(x + y)⋅⎜x + y⋅⎜- ─ - ───────⎟⎟⋅⎜x + y⋅⎜- ─ + ───────⎟⎟
        ⎝      ⎝  2      2   ⎠⎠ ⎝      ⎝  2      2   ⎠⎠

Gröbner bases

Buchberger’s algorithm is implemented, supporting various monomial orders:

>>> groebner([x**2 + 1, y**4*x + x**3])
             ⎛⎡ 2       4    ⎤                           ⎞
GroebnerBasis⎝⎣x  + 1, y  - 1⎦, x, y, domain=ℤ, order=lex⎠


>>> groebner([x**2 + 1, y**4*x + x**3, x*y*z**3], order=grevlex)
             ⎛⎡ 4       3   2    ⎤                                  ⎞
GroebnerBasis⎝⎣y  - 1, z , x  + 1⎦, x, y, z, domain=ℤ, order=grevlex⎠