Integer-valued polynomial rings¶

AUTHORS:

Frédéric Chapoton (2023-03): Initial version

class sage.rings.polynomial.integer_valued_polynomials.IntegerValuedPolynomialRing(R)[source]¶

Bases: UniqueRepresentation, Parent

The integer-valued polynomial ring over a base ring \(R\).

Integer-valued polynomial rings are commutative and associative algebras, with a basis indexed by nonnegative integers.

There are two natural bases, made of the sequence \(\binom{x}{n}\) for \(n \geq 0\) (the binomial basis) and of the other sequence \(\binom{x+n}{n}\) for \(n \geq 0\) (the shifted basis).

These two bases are available as follows:

Sage

sage: B = IntegerValuedPolynomialRing(QQ).Binomial()
sage: S = IntegerValuedPolynomialRing(QQ).Shifted()

Python

>>> from sage.all import *
>>> B = IntegerValuedPolynomialRing(QQ).Binomial()
>>> S = IntegerValuedPolynomialRing(QQ).Shifted()

Sage Live

B = IntegerValuedPolynomialRing(QQ).Binomial()
S = IntegerValuedPolynomialRing(QQ).Shifted()

or by using the shortcuts:

Sage

sage: B = IntegerValuedPolynomialRing(QQ).B()
sage: S = IntegerValuedPolynomialRing(QQ).S()

Python

>>> from sage.all import *
>>> B = IntegerValuedPolynomialRing(QQ).B()
>>> S = IntegerValuedPolynomialRing(QQ).S()

Sage Live

B = IntegerValuedPolynomialRing(QQ).B()
S = IntegerValuedPolynomialRing(QQ).S()

There is a conversion formula between the two bases:

\[\binom{x}{i} = \sum_{k=0}^{i} (-1)^{i-k} \binom{i}{k} \binom{x+k}{k}\]

with inverse:

\[\binom{x+i}{i} = \sum_{k=0}^{i} \binom{i}{k} \binom{x}{k}.\]

REFERENCES:

Wikipedia article Integer-valued polynomial

B[source]¶: alias of Binomial

class Bases(parent_with_realization)[source]¶

Bases: Category_realization_of_parent

class ElementMethods[source]¶

Bases: object

content()[source]¶

Return the content of self.

This is the gcd of the coefficients.

EXAMPLES:

Sage

sage: F = IntegerValuedPolynomialRing(ZZ).S()
sage: B = F.basis()
sage: (3*B[4]+6*B[7]).content()
3

Python

>>> from sage.all import *
>>> F = IntegerValuedPolynomialRing(ZZ).S()
>>> B = F.basis()
>>> (Integer(3)*B[Integer(4)]+Integer(6)*B[Integer(7)]).content()
3

Sage Live

F = IntegerValuedPolynomialRing(ZZ).S()
B = F.basis()
(3*B[4]+6*B[7]).content()

polynomial()[source]¶

Convert to a polynomial in \(x\).

EXAMPLES:

Sage

sage: F = IntegerValuedPolynomialRing(ZZ).S()
sage: B = F.gen()
sage: (B+1).polynomial()
x + 2

sage: F = IntegerValuedPolynomialRing(ZZ).B()
sage: B = F.gen()
sage: (B+1).polynomial()
x + 1

Python

>>> from sage.all import *
>>> F = IntegerValuedPolynomialRing(ZZ).S()
>>> B = F.gen()
>>> (B+Integer(1)).polynomial()
x + 2

>>> F = IntegerValuedPolynomialRing(ZZ).B()
>>> B = F.gen()
>>> (B+Integer(1)).polynomial()
x + 1

Sage Live

F = IntegerValuedPolynomialRing(ZZ).S()
B = F.gen()
(B+1).polynomial()
F = IntegerValuedPolynomialRing(ZZ).B()
B = F.gen()
(B+1).polynomial()

shift(j=1)[source]¶

Shift all indices by \(j\).

INPUT:

j – integer (default: 1)

In the binomial basis, the shift by 1 corresponds to a summation operator from \(0\) to \(x\).

EXAMPLES:

Sage

sage: F = IntegerValuedPolynomialRing(ZZ).B()
sage: B = F.gen()
sage: (B+1).shift()
B[1] + B[2]
sage: (B+1).shift(3)
B[3] + B[4]

Python

>>> from sage.all import *
>>> F = IntegerValuedPolynomialRing(ZZ).B()
>>> B = F.gen()
>>> (B+Integer(1)).shift()
B[1] + B[2]
>>> (B+Integer(1)).shift(Integer(3))
B[3] + B[4]

Sage Live

F = IntegerValuedPolynomialRing(ZZ).B()
B = F.gen()
(B+1).shift()
(B+1).shift(3)

sum_of_coefficients()[source]¶

Return the sum of coefficients.

In the shifted basis, this is the evaluation at \(x=0\).

EXAMPLES:

Sage

sage: F = IntegerValuedPolynomialRing(ZZ).S()
sage: B = F.basis()
sage: (B[2]*B[4]).sum_of_coefficients()
1

Python

>>> from sage.all import *
>>> F = IntegerValuedPolynomialRing(ZZ).S()
>>> B = F.basis()
>>> (B[Integer(2)]*B[Integer(4)]).sum_of_coefficients()
1

Sage Live

F = IntegerValuedPolynomialRing(ZZ).S()
B = F.basis()
(B[2]*B[4]).sum_of_coefficients()

class ParentMethods[source]¶

Bases: object

algebra_generators()[source]¶

Return the generators of this algebra.

EXAMPLES:

Sage

sage: A = IntegerValuedPolynomialRing(ZZ).S(); A
Integer-Valued Polynomial Ring over Integer Ring
in the shifted basis
sage: A.algebra_generators()
Family (S[1],)

Python

>>> from sage.all import *
>>> A = IntegerValuedPolynomialRing(ZZ).S(); A
Integer-Valued Polynomial Ring over Integer Ring
in the shifted basis
>>> A.algebra_generators()
Family (S[1],)

Sage Live

A = IntegerValuedPolynomialRing(ZZ).S(); A
A.algebra_generators()

degree_on_basis(m)[source]¶

Return the degree of the basis element indexed by m.

EXAMPLES:

Sage

sage: A = IntegerValuedPolynomialRing(QQ).S()
sage: A.degree_on_basis(4)
4

Python

>>> from sage.all import *
>>> A = IntegerValuedPolynomialRing(QQ).S()
>>> A.degree_on_basis(Integer(4))
4

Sage Live

A = IntegerValuedPolynomialRing(QQ).S()
A.degree_on_basis(4)

from_polynomial(p)[source]¶

Convert a polynomial into the ring of integer-valued polynomials.

This raises a ValueError if this is not possible.

INPUT:

p – a polynomial in one variable

EXAMPLES:

Sage

sage: A = IntegerValuedPolynomialRing(ZZ).S()
sage: S = A.basis()
sage: S[5].polynomial()
1/120*x^5 + 1/8*x^4 + 17/24*x^3 + 15/8*x^2 + 137/60*x + 1
sage: A.from_polynomial(_)
S[5]
sage: x = polygen(QQ, 'x')
sage: A.from_polynomial(x)
-S[0] + S[1]

sage: A = IntegerValuedPolynomialRing(ZZ).B()
sage: B = A.basis()
sage: B[5].polynomial()
1/120*x^5 - 1/12*x^4 + 7/24*x^3 - 5/12*x^2 + 1/5*x
sage: A.from_polynomial(_)
B[5]
sage: x = polygen(QQ, 'x')
sage: A.from_polynomial(x)
B[1]

Python

>>> from sage.all import *
>>> A = IntegerValuedPolynomialRing(ZZ).S()
>>> S = A.basis()
>>> S[Integer(5)].polynomial()
1/120*x^5 + 1/8*x^4 + 17/24*x^3 + 15/8*x^2 + 137/60*x + 1
>>> A.from_polynomial(_)
S[5]
>>> x = polygen(QQ, 'x')
>>> A.from_polynomial(x)
-S[0] + S[1]

>>> A = IntegerValuedPolynomialRing(ZZ).B()
>>> B = A.basis()
>>> B[Integer(5)].polynomial()
1/120*x^5 - 1/12*x^4 + 7/24*x^3 - 5/12*x^2 + 1/5*x
>>> A.from_polynomial(_)
B[5]
>>> x = polygen(QQ, 'x')
>>> A.from_polynomial(x)
B[1]

Sage Live

A = IntegerValuedPolynomialRing(ZZ).S()
S = A.basis()
S[5].polynomial()
A.from_polynomial(_)
x = polygen(QQ, 'x')
A.from_polynomial(x)
A = IntegerValuedPolynomialRing(ZZ).B()
B = A.basis()
B[5].polynomial()
A.from_polynomial(_)
x = polygen(QQ, 'x')
A.from_polynomial(x)

gen(i=0)[source]¶

Return the generator of this algebra.

The optional argument is ignored.

EXAMPLES:

Sage

sage: F = IntegerValuedPolynomialRing(ZZ).B()
sage: F.gen()
B[1]

Python

>>> from sage.all import *
>>> F = IntegerValuedPolynomialRing(ZZ).B()
>>> F.gen()
B[1]

Sage Live

F = IntegerValuedPolynomialRing(ZZ).B()
F.gen()

gens()[source]¶

Return the generators of this algebra.

EXAMPLES:

Sage

sage: A = IntegerValuedPolynomialRing(ZZ).S(); A
Integer-Valued Polynomial Ring over Integer Ring
in the shifted basis
sage: A.algebra_generators()
Family (S[1],)

Python

>>> from sage.all import *
>>> A = IntegerValuedPolynomialRing(ZZ).S(); A
Integer-Valued Polynomial Ring over Integer Ring
in the shifted basis
>>> A.algebra_generators()
Family (S[1],)

Sage Live

A = IntegerValuedPolynomialRing(ZZ).S(); A
A.algebra_generators()

one_basis()[source]¶

Return the number 0, which index the unit of this algebra.

EXAMPLES:

Sage

sage: A = IntegerValuedPolynomialRing(QQ).S()
sage: A.one_basis()
0
sage: A.one()
S[0]

Python

>>> from sage.all import *
>>> A = IntegerValuedPolynomialRing(QQ).S()
>>> A.one_basis()
0
>>> A.one()
S[0]

Sage Live

A = IntegerValuedPolynomialRing(QQ).S()
A.one_basis()
A.one()

super_categories()[source]¶

Return the super-categories of self.

EXAMPLES:

Sage

sage: A = IntegerValuedPolynomialRing(QQ); A
Integer-Valued Polynomial Ring over Rational Field
sage: C = A.Bases(); C
Category of bases of Integer-Valued Polynomial Ring
over Rational Field
sage: C.super_categories()
[Category of realizations of Integer-Valued Polynomial Ring
 over Rational Field,
 Join of Category of algebras with basis over Rational Field and
 Category of filtered algebras over Rational Field and
 Category of commutative algebras over Rational Field and
 Category of realizations of unital magmas]

Python

>>> from sage.all import *
>>> A = IntegerValuedPolynomialRing(QQ); A
Integer-Valued Polynomial Ring over Rational Field
>>> C = A.Bases(); C
Category of bases of Integer-Valued Polynomial Ring
over Rational Field
>>> C.super_categories()
[Category of realizations of Integer-Valued Polynomial Ring
 over Rational Field,
 Join of Category of algebras with basis over Rational Field and
 Category of filtered algebras over Rational Field and
 Category of commutative algebras over Rational Field and
 Category of realizations of unital magmas]

Sage Live

A = IntegerValuedPolynomialRing(QQ); A
C = A.Bases(); C
C.super_categories()

class Binomial(A)[source]¶

Bases: CombinatorialFreeModule, BindableClass

The integer-valued polynomial ring in the binomial basis.

The basis used here is given by \(B[i] = \binom{x}{i}\) for \(i \in \NN\).

Assuming \(n_1 \leq n_2\), the product of two monomials \(B[n_1] \cdot B[n_2]\) is given by the sum

\[\sum_{k=0}^{n_1} \binom{n_1}{k}\binom{n_1+n_2-k}{n_1} B[n_1 + n_2 - k].\]

The product of two monomials is therefore a positive linear combination of monomials.

EXAMPLES:

Sage

sage: F = IntegerValuedPolynomialRing(QQ).B(); F
Integer-Valued Polynomial Ring over Rational Field
in the binomial basis

sage: F.gen()
B[1]

sage: S = IntegerValuedPolynomialRing(ZZ).B(); S
Integer-Valued Polynomial Ring over Integer Ring
in the binomial basis
sage: S.base_ring()
Integer Ring

sage: G = IntegerValuedPolynomialRing(S).B(); G
Integer-Valued Polynomial Ring over Integer-Valued Polynomial Ring
over Integer Ring in the binomial basis in the binomial basis
sage: G.base_ring()
Integer-Valued Polynomial Ring over Integer Ring
in the binomial basis

Python

>>> from sage.all import *
>>> F = IntegerValuedPolynomialRing(QQ).B(); F
Integer-Valued Polynomial Ring over Rational Field
in the binomial basis

>>> F.gen()
B[1]

>>> S = IntegerValuedPolynomialRing(ZZ).B(); S
Integer-Valued Polynomial Ring over Integer Ring
in the binomial basis
>>> S.base_ring()
Integer Ring

>>> G = IntegerValuedPolynomialRing(S).B(); G
Integer-Valued Polynomial Ring over Integer-Valued Polynomial Ring
over Integer Ring in the binomial basis in the binomial basis
>>> G.base_ring()
Integer-Valued Polynomial Ring over Integer Ring
in the binomial basis

Sage Live

F = IntegerValuedPolynomialRing(QQ).B(); F
F.gen()
S = IntegerValuedPolynomialRing(ZZ).B(); S
S.base_ring()
G = IntegerValuedPolynomialRing(S).B(); G
G.base_ring()

Integer-valued polynomial rings commute with their base ring:

Sage

sage: K = IntegerValuedPolynomialRing(QQ).B()
sage: a = K.gen()
sage: K.is_commutative()
True
sage: L = IntegerValuedPolynomialRing(K).B()
sage: c = L.gen()
sage: L.is_commutative()
True
sage: s = a * c^3; s
B[1]*B[1] + 6*B[1]*B[2] + 6*B[1]*B[3]
sage: parent(s)
Integer-Valued Polynomial Ring over Integer-Valued Polynomial
Ring over Rational Field in the binomial basis in the binomial basis

Python

>>> from sage.all import *
>>> K = IntegerValuedPolynomialRing(QQ).B()
>>> a = K.gen()
>>> K.is_commutative()
True
>>> L = IntegerValuedPolynomialRing(K).B()
>>> c = L.gen()
>>> L.is_commutative()
True
>>> s = a * c**Integer(3); s
B[1]*B[1] + 6*B[1]*B[2] + 6*B[1]*B[3]
>>> parent(s)
Integer-Valued Polynomial Ring over Integer-Valued Polynomial
Ring over Rational Field in the binomial basis in the binomial basis

Sage Live

K = IntegerValuedPolynomialRing(QQ).B()
a = K.gen()
K.is_commutative()
L = IntegerValuedPolynomialRing(K).B()
c = L.gen()
L.is_commutative()
s = a * c^3; s
parent(s)

Integer-valued polynomial rings are commutative:

Sage

sage: c^3 * a == c * a * c * c
True

Python

>>> from sage.all import *
>>> c**Integer(3) * a == c * a * c * c
True

Sage Live

c^3 * a == c * a * c * c

We can also manipulate elements in the basis:

Sage

sage: F = IntegerValuedPolynomialRing(QQ).B()
sage: B = F.basis()
sage: B[2] * B[3]
3*B[3] + 12*B[4] + 10*B[5]
sage: 1 - B[2] * B[2] / 2
B[0] - 1/2*B[2] - 3*B[3] - 3*B[4]

Python

>>> from sage.all import *
>>> F = IntegerValuedPolynomialRing(QQ).B()
>>> B = F.basis()
>>> B[Integer(2)] * B[Integer(3)]
3*B[3] + 12*B[4] + 10*B[5]
>>> Integer(1) - B[Integer(2)] * B[Integer(2)] / Integer(2)
B[0] - 1/2*B[2] - 3*B[3] - 3*B[4]

Sage Live

F = IntegerValuedPolynomialRing(QQ).B()
B = F.basis()
B[2] * B[3]
1 - B[2] * B[2] / 2

and coerce elements from our base field:

Sage

sage: F(4/3)
4/3*B[0]

Python

>>> from sage.all import *
>>> F(Integer(4)/Integer(3))
4/3*B[0]

Sage Live

F(4/3)

class Element[source]¶

Bases: IndexedFreeModuleElement

variable_shift(k=1)[source]¶

Return the image by the shift of variables.

On polynomials, the action is the shift on variables \(x \mapsto x + k\).

INPUT:

k – integer (default: 1)

EXAMPLES:

Sage

sage: A = IntegerValuedPolynomialRing(ZZ).B()
sage: B = A.basis()
sage: B[5].variable_shift()
B[4] + B[5]
sage: B[5].variable_shift(-1)
-B[0] + B[1] - B[2] + B[3] - B[4] + B[5]

Python

>>> from sage.all import *
>>> A = IntegerValuedPolynomialRing(ZZ).B()
>>> B = A.basis()
>>> B[Integer(5)].variable_shift()
B[4] + B[5]
>>> B[Integer(5)].variable_shift(-Integer(1))
-B[0] + B[1] - B[2] + B[3] - B[4] + B[5]

Sage Live

A = IntegerValuedPolynomialRing(ZZ).B()
B = A.basis()
B[5].variable_shift()
B[5].variable_shift(-1)

product_on_basis(n1, n2)[source]¶

Return the product of basis elements n1 and n2.

INPUT:

n1, n2 – integers

EXAMPLES:

Sage

sage: A = IntegerValuedPolynomialRing(QQ).B()
sage: A.product_on_basis(0, 1)
B[1]
sage: A.product_on_basis(1, 2)
2*B[2] + 3*B[3]

Python

>>> from sage.all import *
>>> A = IntegerValuedPolynomialRing(QQ).B()
>>> A.product_on_basis(Integer(0), Integer(1))
B[1]
>>> A.product_on_basis(Integer(1), Integer(2))
2*B[2] + 3*B[3]

Sage Live

A = IntegerValuedPolynomialRing(QQ).B()
A.product_on_basis(0, 1)
A.product_on_basis(1, 2)

S[source]¶: alias of Shifted

class Shifted(A)[source]¶

Bases: CombinatorialFreeModule, BindableClass

The integer-valued polynomial ring in the shifted basis.

The basis used here is given by \(S[i] = \binom{i+x}{i}\) for \(i \in \NN\).

Assuming \(n_1 \leq n_2\), the product of two monomials \(S[n_1] \cdot S[n_2]\) is given by the sum

\[\sum_{k=0}^{n_1} (-1)^k \binom{n_1}{k}\binom{n_1+n_2-k}{n_1} S[n_1 + n_2 - k].\]

EXAMPLES:

Sage

sage: F = IntegerValuedPolynomialRing(QQ).S(); F
Integer-Valued Polynomial Ring over Rational Field
in the shifted basis

sage: F.gen()
S[1]

sage: S = IntegerValuedPolynomialRing(ZZ).S(); S
Integer-Valued Polynomial Ring over Integer Ring
in the shifted basis
sage: S.base_ring()
Integer Ring

sage: G = IntegerValuedPolynomialRing(S).S(); G
Integer-Valued Polynomial Ring over Integer-Valued Polynomial
Ring over Integer Ring in the shifted basis in the shifted basis
sage: G.base_ring()
Integer-Valued Polynomial Ring over Integer Ring
in the shifted basis

Python

>>> from sage.all import *
>>> F = IntegerValuedPolynomialRing(QQ).S(); F
Integer-Valued Polynomial Ring over Rational Field
in the shifted basis

>>> F.gen()
S[1]

>>> S = IntegerValuedPolynomialRing(ZZ).S(); S
Integer-Valued Polynomial Ring over Integer Ring
in the shifted basis
>>> S.base_ring()
Integer Ring

>>> G = IntegerValuedPolynomialRing(S).S(); G
Integer-Valued Polynomial Ring over Integer-Valued Polynomial
Ring over Integer Ring in the shifted basis in the shifted basis
>>> G.base_ring()
Integer-Valued Polynomial Ring over Integer Ring
in the shifted basis

Sage Live

F = IntegerValuedPolynomialRing(QQ).S(); F
F.gen()
S = IntegerValuedPolynomialRing(ZZ).S(); S
S.base_ring()
G = IntegerValuedPolynomialRing(S).S(); G
G.base_ring()

Integer-valued polynomial rings commute with their base ring:

Sage

sage: K = IntegerValuedPolynomialRing(QQ).S()
sage: a = K.gen()
sage: K.is_commutative()
True
sage: L = IntegerValuedPolynomialRing(K).S()
sage: c = L.gen()
sage: L.is_commutative()
True
sage: s = a * c^3; s
S[1]*S[1] + (-6*S[1])*S[2] + 6*S[1]*S[3]
sage: parent(s)
Integer-Valued Polynomial Ring over Integer-Valued Polynomial
Ring over Rational Field in the shifted basis in the shifted basis

Python

>>> from sage.all import *
>>> K = IntegerValuedPolynomialRing(QQ).S()
>>> a = K.gen()
>>> K.is_commutative()
True
>>> L = IntegerValuedPolynomialRing(K).S()
>>> c = L.gen()
>>> L.is_commutative()
True
>>> s = a * c**Integer(3); s
S[1]*S[1] + (-6*S[1])*S[2] + 6*S[1]*S[3]
>>> parent(s)
Integer-Valued Polynomial Ring over Integer-Valued Polynomial
Ring over Rational Field in the shifted basis in the shifted basis

Sage Live

K = IntegerValuedPolynomialRing(QQ).S()
a = K.gen()
K.is_commutative()
L = IntegerValuedPolynomialRing(K).S()
c = L.gen()
L.is_commutative()
s = a * c^3; s
parent(s)

Integer-valued polynomial rings are commutative:

Sage

sage: c^3 * a == c * a * c * c
True

Python

>>> from sage.all import *
>>> c**Integer(3) * a == c * a * c * c
True

Sage Live

c^3 * a == c * a * c * c

We can also manipulate elements in the basis and coerce elements from our base field:

Sage

sage: F = IntegerValuedPolynomialRing(QQ).S()
sage: S = F.basis()
sage: S[2] * S[3]
3*S[3] - 12*S[4] + 10*S[5]
sage: 1 - S[2] * S[2] / 2
S[0] - 1/2*S[2] + 3*S[3] - 3*S[4]

Python

>>> from sage.all import *
>>> F = IntegerValuedPolynomialRing(QQ).S()
>>> S = F.basis()
>>> S[Integer(2)] * S[Integer(3)]
3*S[3] - 12*S[4] + 10*S[5]
>>> Integer(1) - S[Integer(2)] * S[Integer(2)] / Integer(2)
S[0] - 1/2*S[2] + 3*S[3] - 3*S[4]

Sage Live

F = IntegerValuedPolynomialRing(QQ).S()
S = F.basis()
S[2] * S[3]
1 - S[2] * S[2] / 2

class Element[source]¶

Bases: IndexedFreeModuleElement

delta()[source]¶

Return the image by the difference operator \(\Delta\).

The operator \(\Delta\) is defined on polynomials by

\[f \mapsto f(x+1)-f(x).\]

EXAMPLES:

Sage

sage: F = IntegerValuedPolynomialRing(ZZ).S()
sage: S = F.basis()
sage: S[5].delta()
S[0] + S[1] + S[2] + S[3] + S[4]

Python

>>> from sage.all import *
>>> F = IntegerValuedPolynomialRing(ZZ).S()
>>> S = F.basis()
>>> S[Integer(5)].delta()
S[0] + S[1] + S[2] + S[3] + S[4]

Sage Live

F = IntegerValuedPolynomialRing(ZZ).S()
S = F.basis()
S[5].delta()

derivative_at_minus_one()[source]¶

Return the derivative at \(-1\).

This is sometimes useful when \(-1\) is a root.

See also

umbra()

EXAMPLES:

Sage

sage: F = IntegerValuedPolynomialRing(ZZ).S()
sage: B = F.gen()
sage: (B+1).derivative_at_minus_one()
1

Python

>>> from sage.all import *
>>> F = IntegerValuedPolynomialRing(ZZ).S()
>>> B = F.gen()
>>> (B+Integer(1)).derivative_at_minus_one()
1

Sage Live

F = IntegerValuedPolynomialRing(ZZ).S()
B = F.gen()
(B+1).derivative_at_minus_one()

fraction()[source]¶

Return the generating series of values as a fraction.

In the case of Ehrhart polynomials, this is known as the Ehrhart series.

See also

h_vector(), h_polynomial()

EXAMPLES:

Sage

sage: A = IntegerValuedPolynomialRing(ZZ).S()
sage: ex = A.monomial(4)
sage: f = ex.fraction();f
1/(-t^5 + 5*t^4 - 10*t^3 + 10*t^2 - 5*t + 1)

sage: F = LazyPowerSeriesRing(QQ, 't')
sage: F(f)
1 + 5*t + 15*t^2 + 35*t^3 + 70*t^4 + 126*t^5 + 210*t^6 + O(t^7)

sage: poly = ex.polynomial()
sage: [poly(i) for i in range(6)]
[1, 5, 15, 35, 70, 126]

sage: y = polygen(QQ, 'y')
sage: penta = A.from_polynomial(7/2*y^2 + 7/2*y + 1)
sage: penta.fraction()
(t^2 + 5*t + 1)/(-t^3 + 3*t^2 - 3*t + 1)

Python

>>> from sage.all import *
>>> A = IntegerValuedPolynomialRing(ZZ).S()
>>> ex = A.monomial(Integer(4))
>>> f = ex.fraction();f
1/(-t^5 + 5*t^4 - 10*t^3 + 10*t^2 - 5*t + 1)

>>> F = LazyPowerSeriesRing(QQ, 't')
>>> F(f)
1 + 5*t + 15*t^2 + 35*t^3 + 70*t^4 + 126*t^5 + 210*t^6 + O(t^7)

>>> poly = ex.polynomial()
>>> [poly(i) for i in range(Integer(6))]
[1, 5, 15, 35, 70, 126]

>>> y = polygen(QQ, 'y')
>>> penta = A.from_polynomial(Integer(7)/Integer(2)*y**Integer(2) + Integer(7)/Integer(2)*y + Integer(1))
>>> penta.fraction()
(t^2 + 5*t + 1)/(-t^3 + 3*t^2 - 3*t + 1)

Sage Live

A = IntegerValuedPolynomialRing(ZZ).S()
ex = A.monomial(4)
f = ex.fraction();f
F = LazyPowerSeriesRing(QQ, 't')
F(f)
poly = ex.polynomial()
[poly(i) for i in range(6)]
y = polygen(QQ, 'y')
penta = A.from_polynomial(7/2*y^2 + 7/2*y + 1)
penta.fraction()

h_polynomial()[source]¶

Return the \(h\)-vector as a polynomial.

See also

h_vector(), fraction()

EXAMPLES:

Sage

sage: x = polygen(QQ,'x')
sage: A = IntegerValuedPolynomialRing(ZZ).S()
sage: ex = A.from_polynomial((1+x)**3)
sage: ex.h_polynomial()
z^2 + 4*z + 1

Python

>>> from sage.all import *
>>> x = polygen(QQ,'x')
>>> A = IntegerValuedPolynomialRing(ZZ).S()
>>> ex = A.from_polynomial((Integer(1)+x)**Integer(3))
>>> ex.h_polynomial()
z^2 + 4*z + 1

Sage Live

x = polygen(QQ,'x')
A = IntegerValuedPolynomialRing(ZZ).S()
ex = A.from_polynomial((1+x)**3)
ex.h_polynomial()

h_vector()[source]¶

Return the numerator of the generating series of values.

If self is an Ehrhart polynomial, this is the \(h\)-vector.

See also

h_polynomial(), fraction()

EXAMPLES:

Sage

sage: x = polygen(QQ,'x')
sage: A = IntegerValuedPolynomialRing(ZZ).S()
sage: ex = A.from_polynomial((1+x)**3)
sage: ex.h_vector()
(0, 1, 4, 1)

Python

>>> from sage.all import *
>>> x = polygen(QQ,'x')
>>> A = IntegerValuedPolynomialRing(ZZ).S()
>>> ex = A.from_polynomial((Integer(1)+x)**Integer(3))
>>> ex.h_vector()
(0, 1, 4, 1)

Sage Live

x = polygen(QQ,'x')
A = IntegerValuedPolynomialRing(ZZ).S()
ex = A.from_polynomial((1+x)**3)
ex.h_vector()

umbra()[source]¶

Return the Bernoulli umbra.

This is the derivative at \(-1\) of the shift by one.

This is related to Bernoulli numbers.

See also

derivative_at_minus_one()

EXAMPLES:

Sage

sage: F = IntegerValuedPolynomialRing(ZZ).S()
sage: B = F.gen()
sage: (B+1).umbra()
3/2

Python

>>> from sage.all import *
>>> F = IntegerValuedPolynomialRing(ZZ).S()
>>> B = F.gen()
>>> (B+Integer(1)).umbra()
3/2

Sage Live

F = IntegerValuedPolynomialRing(ZZ).S()
B = F.gen()
(B+1).umbra()

variable_shift(k=1)[source]¶

Return the image by the shift of variables.

On polynomials, the action is the shift on variables \(x \mapsto x + k\).

INPUT:

k – integer (default: 1)

EXAMPLES:

Sage

sage: A = IntegerValuedPolynomialRing(ZZ).S()
sage: S = A.basis()
sage: S[5].variable_shift()
S[0] + S[1] + S[2] + S[3] + S[4] + S[5]

sage: S[5].variable_shift(-1)
-S[4] + S[5]

Python

>>> from sage.all import *
>>> A = IntegerValuedPolynomialRing(ZZ).S()
>>> S = A.basis()
>>> S[Integer(5)].variable_shift()
S[0] + S[1] + S[2] + S[3] + S[4] + S[5]

>>> S[Integer(5)].variable_shift(-Integer(1))
-S[4] + S[5]

Sage Live

A = IntegerValuedPolynomialRing(ZZ).S()
S = A.basis()
S[5].variable_shift()
S[5].variable_shift(-1)

from_h_vector(h)[source]¶

Convert from some \(h\)-vector.

INPUT:

h – tuple or vector

See also

Element.h_vector()

EXAMPLES:

Sage

sage: A = IntegerValuedPolynomialRing(ZZ).S()
sage: S = A.basis()
sage: ex = S[2]+S[4]
sage: A.from_h_vector(ex.h_vector())
S[2] + S[4]

Python

>>> from sage.all import *
>>> A = IntegerValuedPolynomialRing(ZZ).S()
>>> S = A.basis()
>>> ex = S[Integer(2)]+S[Integer(4)]
>>> A.from_h_vector(ex.h_vector())
S[2] + S[4]

Sage Live

A = IntegerValuedPolynomialRing(ZZ).S()
S = A.basis()
ex = S[2]+S[4]
A.from_h_vector(ex.h_vector())

product_on_basis(n1, n2)[source]¶

Return the product of basis elements n1 and n2.

INPUT:

n1, n2 – integers

EXAMPLES:

Sage

sage: A = IntegerValuedPolynomialRing(QQ).S()
sage: A.product_on_basis(0, 1)
S[1]
sage: A.product_on_basis(1, 2)
-2*S[2] + 3*S[3]

Python

>>> from sage.all import *
>>> A = IntegerValuedPolynomialRing(QQ).S()
>>> A.product_on_basis(Integer(0), Integer(1))
S[1]
>>> A.product_on_basis(Integer(1), Integer(2))
-2*S[2] + 3*S[3]

Sage Live

A = IntegerValuedPolynomialRing(QQ).S()
A.product_on_basis(0, 1)
A.product_on_basis(1, 2)

a_realization()[source]¶

Return a default realization.

The Binomial realization is chosen.

EXAMPLES:

Sage

sage: IntegerValuedPolynomialRing(QQ).a_realization()
Integer-Valued Polynomial Ring over Rational Field
in the binomial basis

Python

>>> from sage.all import *
>>> IntegerValuedPolynomialRing(QQ).a_realization()
Integer-Valued Polynomial Ring over Rational Field
in the binomial basis

Sage Live

IntegerValuedPolynomialRing(QQ).a_realization()