多项式

在本节中,我们将介绍如何在 Sage 中创建和使用多项式。

一元多项式

创建多项式环有三种方法。

sage: R = PolynomialRing(QQ, 't')
sage: R
Univariate Polynomial Ring in t over Rational Field
>>> from sage.all import *
>>> R = PolynomialRing(QQ, 't')
>>> R
Univariate Polynomial Ring in t over Rational Field

这会创建一个多项式环,并告诉 Sage 在显示时使用字符串 't' 作为不定元。 然而,这并没有定义符号 t,因此你不能用它来输入属于 R 的多项式(例如 \(t^2+1\))。

另一种方法是

sage: S = QQ['t']
sage: S == R
True
>>> from sage.all import *
>>> S = QQ['t']
>>> S == R
True

这样做对于 t 也存在同样的问题。

第三种非常方便的方法是

sage: R.<t> = PolynomialRing(QQ)
>>> from sage.all import *
>>> R = PolynomialRing(QQ, names=('t',)); (t,) = R._first_ngens(1)

sage: R.<t> = QQ['t']
>>> from sage.all import *
>>> R = QQ['t']; (t,) = R._first_ngens(1)

甚至

sage: R.<t> = QQ[]
>>> from sage.all import *
>>> R = QQ['t']; (t,) = R._first_ngens(1)

这样做还有一个额外的好处,即它定义了变量 t 作为多项式环的不定元, 因此你可以轻松地构造 R 的元素,如下所示。 (请注意,第三种方法与 Magma 中的构造符号非常相似,并且可以像在 Magma 中一样用于广泛的对象。)

sage: poly = (t+1) * (t+2); poly
t^2 + 3*t + 2
sage: poly in R
True
>>> from sage.all import *
>>> poly = (t+Integer(1)) * (t+Integer(2)); poly
t^2 + 3*t + 2
>>> poly in R
True

无论你使用哪种方法定义多项式环,你都可以通过 \(0^{th}\) 生成器恢复不定元:

sage: R = PolynomialRing(QQ, 't')
sage: t = R.0
sage: t in R
True
>>> from sage.all import *
>>> R = PolynomialRing(QQ, 't')
>>> t = R.gen(0)
>>> t in R
True

请注意,类似的构造方法适用于复数:复数可以被视为由符号 i 在实数上生成的,因此我们有以下内容:

sage: CC
Complex Field with 53 bits of precision
sage: CC.0  # 0th generator of CC
1.00000000000000*I
>>> from sage.all import *
>>> CC
Complex Field with 53 bits of precision
>>> CC.gen(0)  # 0th generator of CC
1.00000000000000*I

对于多项式环,你可以在创建环时同时获得环及其生成器,或者仅获得生成器,如下所示:

sage: R, t = QQ['t'].objgen()
sage: t    = QQ['t'].gen()
sage: R, t = objgen(QQ['t'])
sage: t    = gen(QQ['t'])
>>> from sage.all import *
>>> R, t = QQ['t'].objgen()
>>> t    = QQ['t'].gen()
>>> R, t = objgen(QQ['t'])
>>> t    = gen(QQ['t'])

最后我们在 \(\QQ[t]\) 中进行一些算术运算。

sage: R, t = QQ['t'].objgen()
sage: f = 2*t^7 + 3*t^2 - 15/19
sage: f^2
4*t^14 + 12*t^9 - 60/19*t^7 + 9*t^4 - 90/19*t^2 + 225/361
sage: cyclo = R.cyclotomic_polynomial(7); cyclo
t^6 + t^5 + t^4 + t^3 + t^2 + t + 1
sage: g = 7 * cyclo * t^5 * (t^5 + 10*t + 2)
sage: g
7*t^16 + 7*t^15 + 7*t^14 + 7*t^13 + 77*t^12 + 91*t^11 + 91*t^10 + 84*t^9
       + 84*t^8 + 84*t^7 + 84*t^6 + 14*t^5
sage: F = factor(g); F
(7) * t^5 * (t^5 + 10*t + 2) * (t^6 + t^5 + t^4 + t^3 + t^2 + t + 1)
sage: F.unit()
7
sage: list(F)
[(t, 5), (t^5 + 10*t + 2, 1), (t^6 + t^5 + t^4 + t^3 + t^2 + t + 1, 1)]
>>> from sage.all import *
>>> R, t = QQ['t'].objgen()
>>> f = Integer(2)*t**Integer(7) + Integer(3)*t**Integer(2) - Integer(15)/Integer(19)
>>> f**Integer(2)
4*t^14 + 12*t^9 - 60/19*t^7 + 9*t^4 - 90/19*t^2 + 225/361
>>> cyclo = R.cyclotomic_polynomial(Integer(7)); cyclo
t^6 + t^5 + t^4 + t^3 + t^2 + t + 1
>>> g = Integer(7) * cyclo * t**Integer(5) * (t**Integer(5) + Integer(10)*t + Integer(2))
>>> g
7*t^16 + 7*t^15 + 7*t^14 + 7*t^13 + 77*t^12 + 91*t^11 + 91*t^10 + 84*t^9
       + 84*t^8 + 84*t^7 + 84*t^6 + 14*t^5
>>> F = factor(g); F
(7) * t^5 * (t^5 + 10*t + 2) * (t^6 + t^5 + t^4 + t^3 + t^2 + t + 1)
>>> F.unit()
7
>>> list(F)
[(t, 5), (t^5 + 10*t + 2, 1), (t^6 + t^5 + t^4 + t^3 + t^2 + t + 1, 1)]

注意,因式分解正确考虑并记录了单位部分。

如果你在某个研究项目中大量使用某个函数,例如 R.cyclotomic_polynomial, 除了引用 Sage 之外,你还应该尝试找出 Sage 的哪个组件在实际计算分圆多项式并引用它。 在这种情况下,如果你输入 R.cyclotomic_polynomial?? 查看源代码, 你很快会看到一行 f = pari.polcyclo(n),这意味着 PARI 被用于计算分圆多项式。 你的作品中也需要引用 PARI。

除以两个多项式会构造分数域的元素(Sage 会自动创建)。

sage: x = QQ['x'].0
sage: f = x^3 + 1; g = x^2 - 17
sage: h = f/g;  h
(x^3 + 1)/(x^2 - 17)
sage: h.parent()
Fraction Field of Univariate Polynomial Ring in x over Rational Field
>>> from sage.all import *
>>> x = QQ['x'].gen(0)
>>> f = x**Integer(3) + Integer(1); g = x**Integer(2) - Integer(17)
>>> h = f/g;  h
(x^3 + 1)/(x^2 - 17)
>>> h.parent()
Fraction Field of Univariate Polynomial Ring in x over Rational Field

使用 Laurent 级数,可以在 QQ[x] 的分数域中计算级数展开:

sage: R.<x> = LaurentSeriesRing(QQ); R
Laurent Series Ring in x over Rational Field
sage: 1/(1-x) + O(x^10)
1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + O(x^10)
>>> from sage.all import *
>>> R = LaurentSeriesRing(QQ, names=('x',)); (x,) = R._first_ngens(1); R
Laurent Series Ring in x over Rational Field
>>> Integer(1)/(Integer(1)-x) + O(x**Integer(10))
1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + O(x^10)

如果我们给变量不同的命名,我们会得到不同的一元多项式环。

sage: R.<x> = PolynomialRing(QQ)
sage: S.<y> = PolynomialRing(QQ)
sage: x == y
False
sage: R == S
False
sage: R(y)
x
sage: R(y^2 - 17)
x^2 - 17
>>> from sage.all import *
>>> R = PolynomialRing(QQ, names=('x',)); (x,) = R._first_ngens(1)
>>> S = PolynomialRing(QQ, names=('y',)); (y,) = S._first_ngens(1)
>>> x == y
False
>>> R == S
False
>>> R(y)
x
>>> R(y**Integer(2) - Integer(17))
x^2 - 17

环由变量决定。请注意,使用名为 x 的变量创建另一个环不会返回不同的环。

sage: R = PolynomialRing(QQ, "x")
sage: T = PolynomialRing(QQ, "x")
sage: R == T
True
sage: R is T
True
sage: R.0 == T.0
True
>>> from sage.all import *
>>> R = PolynomialRing(QQ, "x")
>>> T = PolynomialRing(QQ, "x")
>>> R == T
True
>>> R is T
True
>>> R.gen(0) == T.gen(0)
True

Sage 还支持任意基环上的幂级数和 Laurent 级数环。 在下面的示例中,我们创建了 \(\GF{7}[[T]]\) 的一个元素, 并通过相除创建 \(\GF{7}((T))\) 的一个元素。

sage: R.<T> = PowerSeriesRing(GF(7)); R
Power Series Ring in T over Finite Field of size 7
sage: f = T  + 3*T^2 + T^3 + O(T^4)
sage: f^3
T^3 + 2*T^4 + 2*T^5 + O(T^6)
sage: 1/f
T^-1 + 4 + T + O(T^2)
sage: parent(1/f)
Laurent Series Ring in T over Finite Field of size 7
>>> from sage.all import *
>>> R = PowerSeriesRing(GF(Integer(7)), names=('T',)); (T,) = R._first_ngens(1); R
Power Series Ring in T over Finite Field of size 7
>>> f = T  + Integer(3)*T**Integer(2) + T**Integer(3) + O(T**Integer(4))
>>> f**Integer(3)
T^3 + 2*T^4 + 2*T^5 + O(T^6)
>>> Integer(1)/f
T^-1 + 4 + T + O(T^2)
>>> parent(Integer(1)/f)
Laurent Series Ring in T over Finite Field of size 7

你也可以使用双括号简写来创建幂级数环:

sage: GF(7)[['T']]
Power Series Ring in T over Finite Field of size 7
>>> from sage.all import *
>>> GF(Integer(7))[['T']]
Power Series Ring in T over Finite Field of size 7

多元多项式

要处理多个变量的多项式,我们首先声明多项式环和变量。

sage: R = PolynomialRing(GF(5),3,"z") # here, 3 = number of variables
sage: R
Multivariate Polynomial Ring in z0, z1, z2 over Finite Field of size 5
>>> from sage.all import *
>>> R = PolynomialRing(GF(Integer(5)),Integer(3),"z") # here, 3 = number of variables
>>> R
Multivariate Polynomial Ring in z0, z1, z2 over Finite Field of size 5

与定义一元多项式环一样,有多种方法:

sage: GF(5)['z0, z1, z2']
Multivariate Polynomial Ring in z0, z1, z2 over Finite Field of size 5
sage: R.<z0,z1,z2> = GF(5)[]; R
Multivariate Polynomial Ring in z0, z1, z2 over Finite Field of size 5
>>> from sage.all import *
>>> GF(Integer(5))['z0, z1, z2']
Multivariate Polynomial Ring in z0, z1, z2 over Finite Field of size 5
>>> R = GF(Integer(5))['z0, z1, z2']; (z0, z1, z2,) = R._first_ngens(3); R
Multivariate Polynomial Ring in z0, z1, z2 over Finite Field of size 5

此外,如果你想让变量名为单个字母,你可以使用以下简写:

sage: PolynomialRing(GF(5), 3, 'xyz')
Multivariate Polynomial Ring in x, y, z over Finite Field of size 5
>>> from sage.all import *
>>> PolynomialRing(GF(Integer(5)), Integer(3), 'xyz')
Multivariate Polynomial Ring in x, y, z over Finite Field of size 5

接下来让我们进行一些算术运算。

sage: z = GF(5)['z0, z1, z2'].gens()
sage: z
(z0, z1, z2)
sage: (z[0]+z[1]+z[2])^2
z0^2 + 2*z0*z1 + z1^2 + 2*z0*z2 + 2*z1*z2 + z2^2
>>> from sage.all import *
>>> z = GF(Integer(5))['z0, z1, z2'].gens()
>>> z
(z0, z1, z2)
>>> (z[Integer(0)]+z[Integer(1)]+z[Integer(2)])**Integer(2)
z0^2 + 2*z0*z1 + z1^2 + 2*z0*z2 + 2*z1*z2 + z2^2

你还可以使用更多数学符号来构造多项式环。

sage: R = GF(5)['x,y,z']
sage: x,y,z = R.gens()
sage: QQ['x']
Univariate Polynomial Ring in x over Rational Field
sage: QQ['x,y'].gens()
(x, y)
sage: QQ['x'].objgens()
(Univariate Polynomial Ring in x over Rational Field, (x,))
>>> from sage.all import *
>>> R = GF(Integer(5))['x,y,z']
>>> x,y,z = R.gens()
>>> QQ['x']
Univariate Polynomial Ring in x over Rational Field
>>> QQ['x,y'].gens()
(x, y)
>>> QQ['x'].objgens()
(Univariate Polynomial Ring in x over Rational Field, (x,))

多元多项式在 Sage 中使用 Python 字典和多项式的“分配表示”实现。 Sage 使用了一些 Singular [Si] ,例如,用于计算理想的最大公约数和 Gröbner 基。

sage: R, (x, y) = PolynomialRing(RationalField(), 2, 'xy').objgens()
sage: f = (x^3 + 2*y^2*x)^2
sage: g = x^2*y^2
sage: f.gcd(g)
x^2
>>> from sage.all import *
>>> R, (x, y) = PolynomialRing(RationalField(), Integer(2), 'xy').objgens()
>>> f = (x**Integer(3) + Integer(2)*y**Integer(2)*x)**Integer(2)
>>> g = x**Integer(2)*y**Integer(2)
>>> f.gcd(g)
x^2

接下来我们通过简单地将 (f,g) 乘以 R 来创建由 \(f\)\(g\) 生成的理想 \((f,g)\),(也可以写做 ideal([f,g])ideal(f,g))。

sage: I = (f, g)*R; I
Ideal (x^6 + 4*x^4*y^2 + 4*x^2*y^4, x^2*y^2) of Multivariate Polynomial
Ring in x, y over Rational Field
sage: B = I.groebner_basis(); B
[x^6, x^2*y^2]
sage: x^2 in I
False
>>> from sage.all import *
>>> I = (f, g)*R; I
Ideal (x^6 + 4*x^4*y^2 + 4*x^2*y^4, x^2*y^2) of Multivariate Polynomial
Ring in x, y over Rational Field
>>> B = I.groebner_basis(); B
[x^6, x^2*y^2]
>>> x**Integer(2) in I
False

顺便说一句,上面的 Gröbner 基不是一个列表,而是一个不可变序列。 这意味着它有全集,父结构,并且不可更改(这是好的,因为更改基会破坏使用 Gröbner 基的其他例程)。

sage: B.universe()
Multivariate Polynomial Ring in x, y over Rational Field
sage: B[1] = x
Traceback (most recent call last):
...
ValueError: object is immutable; please change a copy instead.
>>> from sage.all import *
>>> B.universe()
Multivariate Polynomial Ring in x, y over Rational Field
>>> B[Integer(1)] = x
Traceback (most recent call last):
...
ValueError: object is immutable; please change a copy instead.

Sage 中有一些(没有我们想要的那么多)交换代数可用,通过 Singular 实现。 例如,我们可以计算 \(I\) 的初等分解和相关素数:

sage: I.primary_decomposition()
[Ideal (x^2) of Multivariate Polynomial Ring in x, y over Rational Field,
 Ideal (y^2, x^6) of Multivariate Polynomial Ring in x, y over Rational Field]
sage: I.associated_primes()
[Ideal (x) of Multivariate Polynomial Ring in x, y over Rational Field,
 Ideal (y, x) of Multivariate Polynomial Ring in x, y over Rational Field]
>>> from sage.all import *
>>> I.primary_decomposition()
[Ideal (x^2) of Multivariate Polynomial Ring in x, y over Rational Field,
 Ideal (y^2, x^6) of Multivariate Polynomial Ring in x, y over Rational Field]
>>> I.associated_primes()
[Ideal (x) of Multivariate Polynomial Ring in x, y over Rational Field,
 Ideal (y, x) of Multivariate Polynomial Ring in x, y over Rational Field]