def poly_discriminant(p, symbol):
"""
Returns the discriminant of a polynomial with respect to symbol.
The discriminant of a univariate polynomial p of degree n is defined as
n**(n*(n-1)/2)/a_n*resultant(p, p'), where p' is the derivative of p and a_n
is the leading coefficient of p. Because the resultant of two polynomials
vanishes identically whenever the two polynomials share a root, and a
polynomial shares a root with its derivative if and only if the root is a
repeated root, it follows that the discriminant of a polynomial vanishes
identically if and only if the polynomial has a repeated root.
See also:
<http://en.wikipedia.org/wiki/Discriminant>
Example:
>>> from sympy import discriminant, Poly
>>> from sympy.abc import a, b, c, x
>>> discriminant(Poly(a*x**2 + b*x + c, x), x)
-4*a*c + b**2
>>> discriminant(Poly(2*x**5 + x**2 + 10, x), x)
500004320
>>> discriminant(Poly((x-1)*(x+1), x), x)
4
>>> discriminant(Poly((x-1)**2*(x+1), x), x)
0
"""
if not isinstance(p, Poly):
p = Poly(p, symbol)
if not p.is_univariate:
# We need to make p univariate for poly_resultant to work
p = Poly(p.as_basic(), symbol) # Is there a better way to do this?
if p.degree == 0:
return S.Zero
return (S(int((-1)**((p.degree)*(S(p.degree) - 1)/2)))/p.lead_coeff*\
poly_resultant(p, p.diff(symbol))).expand()
def poly_sqf(f, *symbols):
"""Compute square-free decomposition of an univariate polynomial.
Given an univariate polynomial f over an unique factorization domain
returns tuple (f_1, f_2, ..., f_n), where all A_i are co-prime and
square-free polynomials and f = f_1 * f_2**2 * ... * f_n**n.
>>> from sympy import *
>>> x,y = symbols('xy')
>>> p, q = poly_sqf(x*(x+1)**2, x)
>>> p.as_basic()
x
>>> q.as_basic()
1 + x
For more information on the implemented algorithm refer to:
[1] M. Bronstein, Symbolic Integration I: Transcendental
Functions, Second Edition, Springer-Verlang, 2005
[2] J. von zur Gathen, J. Gerhard, Modern Computer Algebra,
Second Edition, Cambridge University Press, 2003
"""
if not isinstance(f, Poly):
f = Poly(f, *symbols)
elif symbols:
raise SymbolsError("Redundant symbols were given")
if f.is_multivariate:
raise MultivariatePolyError(f)
coeff, f = f.as_primitive()
sqf = []
h = f.diff()
g = poly_gcd(f, h)
p = poly_div(f, g)[0]
q = poly_div(h, g)[0]
p, q = poly_reduce(p, q)
while True:
h = q - p.diff()
if h.is_zero:
break
g = poly_gcd(p, h)
sqf.append(g)
p = poly_div(p, g)[0]
q = poly_div(h, g)[0]
p, q = poly_reduce(p, q)
sqf.append(p)
head, tail = sqf[0], sqf[1:]
head = head.mul_term(coeff)
return [head] + tail
def poly_sqf(f, *symbols):
"""Compute square-free decomposition of an univariate polynomial.
Given an univariate polynomial f over an unique factorization domain
returns tuple (f_1, f_2, ..., f_n), where all A_i are co-prime and
square-free polynomials and f = f_1 * f_2**2 * ... * f_n**n.
>>> from sympy import *
>>> x,y = symbols('xy')
>>> p, q = poly_sqf(x*(x+1)**2, x)
>>> p.as_basic()
x
>>> q.as_basic()
1 + x
For more information on the implemented algorithm refer to:
[1] M. Bronstein, Symbolic Integration I: Transcendental
Functions, Second Edition, Springer-Verlang, 2005
[2] J. von zur Gathen, J. Gerhard, Modern Computer Algebra,
Second Edition, Cambridge University Press, 2003
"""
if not isinstance(f, Poly):
f = Poly(f, *symbols)
elif symbols:
raise SymbolsError("Redundant symbols were given")
if f.is_multivariate:
raise MultivariatePolyError(f)
coeff, f = f.as_primitive()
sqf = []
h = f.diff()
g = poly_gcd(f, h)
p = poly_div(f, g)[0]
q = poly_div(h, g)[0]
p, q = poly_reduce(p, q)
while True:
h = q - p.diff()
if h.is_zero:
break
g = poly_gcd(p, h)
sqf.append(g)
p = poly_div(p, g)[0]
q = poly_div(h, g)[0]
p, q = poly_reduce(p, q)
sqf.append(p)
head, tail = sqf[0], sqf[1:]
head = head.mul_term(coeff)
return [head] + tail
